Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1

Page 16 Exercise 1 Answer

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to find an equation of that shows how much each of them still owes and the other question to consider is have the problem been solved before.

The amount that each still owes can be found by defining a variable to unknown and use the known quantities.

Find the equation of that each owes.

Nori, her friend, and her mother bought a baseball game ticket package.

The package includes good seats, lunch, and a chance to get autographs from players.

The total cost for the three of them is $375.

They each paid a $50 deposit.

Total cost is$375

Each paid $50 deposit.

Define a variable.
​x  =  $50
y  =  $50
z  =  $50

Total amount paid as deposit is,

x + y + z = $50 + $50 + $50

= $150

Find the total amount that has to be pay.

Total cost of the ticket is$375

= $375 − $150

=$225

Find the amount that each still owes.

Total amount they owe divided by 3.

= $225 ÷ 3

Page 16 Exercise 1 Answer
= $75

The amount that each still owes is $75.

I have solved similar problems before.

The final answer is that each still owes $75 and the similar problem has been solved before.

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1

Envision Math Grade 8 Standards For Mathematical Practice Exercise 1 Solutions

Page 16 Exercise 2 Answer

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to find that what information are necessary and what are not necessary.

The necessary and unnecessary information can be found by observing the problem.

Find what information are necessary and what are not necessary.

Nori, her friend, and her mother bought a baseball game ticket package.

The package includes good seats, lunch, and a chance to get autographs from players.

The total cost for the three of them is $375.

They each paid a $50 deposit.

Necessary information:

The total cost for the three of them is $375.

They each paid a $50 deposit.

Unnecessary information:

The package includes good seats, lunch, and a chance to get autographs from players

The necessary information is that the total cost for the three of them is$375 and they each paid a $50 deposit.

The unnecessary information is the package includes good seats, lunch, and a chance to get autographs from players.

 

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 16 Exercise 3 Answer

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to check whether the answer make sense.

The answer whether it makes sense or not can be found using the results we obtained.

Check whether the answer make sense.

Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The answer found is that the amount that each still owes is $75.

Yes, it makes send because total amount paid is $375.

For each of them the price of the ticket will be $150

Subtract total amount paid minus total deposit amount.

= $375 − $150

= $225

Divide the amount into three,

= $ 225÷3
Page 16 Exercise 3 Answer
= $ 75

Yes, the answer makes sense.

Envision Math Grade 8 Volume 1 Standards For Mathematical Practice Answers

Page 16 Exercise 4 Answer

Given Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The objective is to find whether the solution pathway is same as the classmate.

The amount that each still owes can be found by defining a variable to unknown and use the known quantities.

Find whether the solution pathway is same as the classmate.
Nori, her friend, and her mother bought a baseball game ticket package. The package includes good seats, lunch, and a chance to get autographs from players. The total cost for the three of them is $375.

The solution pathway is,

Total cost is $375

Each paid $50 deposit.

Define a variable.
​x = $50
y = $50
z = $50

Total amount paid as deposit is,
x + y + z = $50 + $50 + $50

= $150

Find the total amount that has to be pay.

Total cost of the ticket is $375

= $375 − $150

= $225

Find the amount that each still owes.

Total amount they owe divided by3.

​= $225 ÷ 3
Page 16 Exercise 4 Answer

= $75

The amount that each still owes is $75.

The solution pathway is same for me as well as for my classmate.

Envision Math Grade 8 Student Edition Solutions For Standards For Mathematical Practice Exercise 1

Page 18 Exercise 2 Answer

Given Michael’s class is conducting an experiment by tossing a coin. Tails has come up 5 times in a row. That means the next toss will land heads up.

The objective is to find what arguments can be present to defend the conjecture.

The conjecture can be found using the information provided.

Find what arguments can be present to defend the conjecture.

Michael’s class is conducting an experiment by tossing a coin.

The table below shows the results of the last tosses.

Page 18 Exercise 2 Answer

The two outcomes of the toss of a coin are heads or tails. For any individual toss of the coin, the outcome will be either heads o. tails. The two outcomes (heads or tails) are therefore mutually exclusive; if the coin comes up heads on a single toss, it cannot come up tails on the same toss.

The arguments that can be present to defend the conjecture is,

A coin can land on heads or tails, so there are two equally likely outcomes.

Because there may be either head or tail.

The arguments that can be present to defend the conjecture is,

A coin can land on heads or tails, so there are two equally likely outcomes.

Solutions For Envision Math Grade 8 Exercise 1 Standards For Mathematical Practice

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 18 Exercise 3 Answer

Given Michael’s class is conducting an experiment by tossing a coin. The table below shows the results of the last 9 tosses.

The objective is to find what conjecture can be made about the solution to the problem.

The conjecture can be found using the information provided.

Find what conjecture can be made about the solution to the problem.

Michael’s class is conducting an experiment by tossing a coin.

The table below shows the results of the last 9 tosses.

Page 18 Exercise 3 Answer

 

The conjecture that can be made about the solution to the problem is,

Total observations=25

Required outcome Tails {T,T,T,T,T}

This can occur only ONCE!

The conjecture made is as there are continuous five tails, this may end up with head at next toss.

Envision Math Exercise 1 Solutions For Grade 8 Standards Practice

Page 19 Exercise 1 Answer

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the representation that is used to show the relationship among quantities or variables.

The representation can be found using the modelling capability.

Find the representation that is used to show the relationship among quantities or variables.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The possible outcomes are, HH, TT, TH, & HT

Page 19 Exercise 1 Answer Image 1

Representation of the outcomes is,

The representation of the relationship among the quantities is,

Page 19 Exercise 1 Answer Image 2

Page 19 Exercise 2 Answer

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the way to make our model to work better.

The representation can be found using the modelling capability.

Find the way to make our model to work better.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more

likely that the coins will show one heads and one tails.

To make the model better if it doesn’t work, use table representation.

Page 19 Exercise 2 Answer

If the model doesn’t work, the choose better model like table representation

Page 19 Exercise 2 Answer

Page 19 Exercise 3 Answer

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find the assumption that can be made to simplify the problem.

The simplified problem can be found using the better assumption.

Determine the assumption that can be made to simplify the problem.

The class decided to toss two coins at the same time.

They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

When we toss two coins simultaneously then the possible of outcomes are:

(Two heads) or (one head and one tail) or (two tails) i.e., in short (H,H)(H,T)(T,T) respectively;

Where H is denoted for head and T is denoted for tail.

The assumption made to simplify the problem is when two coins is tossed simultaneously the possible of outcomes are: (Two heads) or (one head and one tail) or (two tails)

Standards For Mathematical Practice Envision Math Grade 8 Solutions Exercise 1

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 19 Exercise 4 Answer

 

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to check whether the solution make sense.

The simplified problem can be found using the better assumption.

Check whether the solution make sense.

The class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The solution obtained is,

The possible outcomes are,

HH, TT, TH, & HT

Page 19 Exercise 4 Answer

 

The solution makes sense.

Yes. The solution or prediction we had makes sense. Because on flipping two coins the outcomes should be both heads or both tails or one head and one tail.

Practice Problems For Envision Math Grade 8 Standards Exercise 1

Page 19 Exercise 5 Answer

Given the class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

The objective is to find is there something that is forgotten or not considered.

The representation can be found using the modelling capability.

Find is there something that is forgotten or not considered.

The class decided to toss two coins at the same time. They wanted to decide whether it is more likely that both coins will show the same side, heads-heads or tails-tails, or more likely that the coins will show one heads and one tails.

There is nothing that is forgotten or not considered while predicting the solution.

Nothing is forgotten or not considered while predicting the solution.

Envision Math Grade 8 Chapter Standards Practice Exercise 1 Worksheet Answers

Page 20 Exercise 1 Answer

Given The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about 1.618 times longer than the shorter side. Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.

The objective is to find whether different tools can be used.

The tools that are to be used can be found by strategical thinking.

Find whether different tools can be used.
The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side.

Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.
Large advertising signs.
Page 20 Exercise 1 Answer Image 1
Medium advertising signs.
Page 20 Exercise 1 Answer Image 2

Small advertising signs.
Page 20 Exercise 1 Answer Image 3

Different tools can be used to make design and technology software’s for drawing.

 

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 20 Exercise 2 Answer

Given The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side. Draw rectangles to scale to create templates of possible small, medium, and large advertising signs.

The objective is to find what other resources can uses to reach the solution

The tools that are to be used can be found by strategical thinking.

Find what other resources can uses to reach the solution.
The Golden Company uses signs in the shape of golden rectangles to advertise its products. In a golden rectangle the length of the longer side is about times longer than the shorter side.

The other resources that can be used to help reach the solution is technical drawing software where the measurements will be given and content to be advertise will also be given.

The technical drawing software is the resource that is used to reach the solution.

 

Page 21 Exercise 1 Answer

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to state the meaning of variables and symbols used.

The variables and symbols used can be found using the information provided

State the meaning of variables and symbols used.

The work is precise enough and appropriate.

Draw the shorter side and then multiply that dimension by 1.618 to determine the length of the longer side.

Length L = Shorter side x1.618
Width W =1.618 cm

The variables used are,

L for length of the rectangle.

W for width of the rectangle.

Symbols that are used is,

Equal to =

Multiplication x

The symbols used are equal to and multiplication.

The variables used are L and W.

 

Page 21 Exercise 2 Answer

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to specify the units of measure using.

The variables and symbols used can be found using the information provided.

State the meaning of variables and symbols used.

The work is precise enough and appropriate.

Draw the shorter side and then multiply that dimension by 1.618 to determine the length of the longer side.

Length L = Shorter side x1.618

Width W =1.618cm

The variables used are,
L for length of the rectangle.
W for width of the rectangle.
Symbols that are used is,
Equal to =

Multiplication x

The units of measure that is specified is centimetre or cm.

The units of measure that is specified is centimetre or cm.

 

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 21 Exercise 3 Answer

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to find whether the work is precise or exact enough.

The precision used can be found using the result got.

Find whether the work is precise or exact enough.
Page 21 Exercise 3 Answer

I am using the appropriate tools to make sure that the dimensions of the templates are precise.

The work is precise and exact because the appropriate tools are used.

 

Page 21 Exercise 4 Answer

Given the explanation about golden rectangle advertisement and their dimensions.

The objective is to find whether the explanation carefully formulated.

The variables and symbols used can be found using the information provided.

Find whether the explanation carefully formulated.

Length L = Shorter side x1.618
Width W = 1.618cm

The variables used are,

L for length of the rectangle.

W for width of the rectangle.

Symbols that are used is,

Equal to =

Multiplication x1

The units of measure that is specified are centimetre or cm.

Yes, the information provided are carefully formulated.

Yes, the explanations provided are carefully formulated.

 

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 22 Exercise 1 Answer

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to make a chart that shows drawings of the cell divisions through 10 divisions.

The chart can be found using the table and the explanation provided.

Make a chart that shows drawings of the cell divisions through 10 divisions.

The table below shows the number of cells after a certain number of divisions.
Page 22 Exercise 1 Answer Image 1
The structure for the problem is,
1 cell becomes 2 cells and 2 cells become 4 cells, and so on.
Page 22 Exercise 1 Answer Image 2

The attributes that are used in the drawing is a tree chart to describe the cells and their division.

 

Page 22 Exercise 2 Answer

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to find the patterns in numbers that are described.

The patterns in number can be found using observation.

Find the patterns in numbers that are described.

Stuart is studying cell division. The table shows the number of cells after a certain number of divisions.

Initial cell is one and on further cell division, it becomes 2.

2 cells become 4

4 cells become 8 and so on….

1, 2, 4, 8, 16, 32, 64, 128, 256,….

This sequence has a factor of 2 between each number.

Each term (except this first term) is found by multiplying the previous term by 2.

The pattern obtained in a geometric sequence.

The pattern in numbers can be described as GEOMETRIC SEQUENCE.

 

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 22 Exercise 3 Answer

Given that Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

The objective is to find th expressions or equations in different ways.

The patterns in number can be found using observation.

Find the expressions or equations in different ways.

Stuart is studying cell division. The table below shows the number of cells after a certain number of divisions.

1, 2, 4, 8, 16, 32, 64, 128, 256,…..

This sequence has a factor of 2 between each number.

Each term (except the first term) is found by multiplying the previous term by 2.

The pattern obtained in geometric sequence.

The equation of expression that can is noticed is 2n.

Example:

Initial cell is one and on further cell division, it becomes 2.

On second division is should become 4.

For that multiply the cells obtained in initial division by 2.

The expression or equation can be viewed in different ways. The expression we obtained is 2n.

How To Solve Envision Math Grade 8 Standards For Mathematical Practice Exercise 1

Page 23 Exercise 1 Answer

Given the explanation about cell division.

Page 23 Exercise 1 Answer Image 1

The objective is to find the general method or shortcut to solve the problem.

The general methods can be found using simple formulas and observing.

Find the general method or shortcut to solve the problem.

From the figure, it is observed that the cells are divided from one cell.

The general method that can be used to solve the problem is,

Number of cells multiplied by 2 is the result.

It is noticed that when calculations are repeated.

Then we can find more general methods and shortcuts.
Page 23 Exercise 1 Answer Image 2

The final answer is that it can be generalized and the shortcut used is power tables as the cells divided is multiplied by two will give the result.

Page 23 Exercise 2 Answer

Given the explanation about cell division.

The objective is to find that an expression or equation can be derived.

The general methods can be found using the simple formulas and observation.

Find an expression or equation that can be derived from the examples.

The example obtained is,
Page 23 Exercise 2 Answer

The expression or equation that can be derived is,

N = n x 2

Equation:

4 * 2 = 8

Where N = total number of cells.

n = number of cells divided.

The final answer is that an expression that can be derived is n N = n x 2

Envision Math Grade 8 Standards For Mathematical Practice Exercise 1 Explained

Envision Math Grade 8 Volume 1 Student Edition Solutions chapter Standards for Mathematical Practice Exercise 1 Page 23 Exercise 3 Answer

Given the explanation about cell division.
Page 23 Exercise 3 Answer

The objective is to find that how reasonable are the results.

The general methods can be found using the simple formula and observation.

Find that how reasonable are the results.

The expression or equation that can be derived is,

Where N = total number of cells.
n = number of cells divided.

The result obtained is so reasonable that, using expression we obtained, number of cells that are divided can be found out easily without drawing 1024 cells in a chat that takes time hardly.

The final answer is that, the result we are getting is so reasonable that using the formula we obtained, it can be used to solve the cell divide.

Envision Math Grade 8 Chapter 1 Real Numbers Topic 1 Solutions

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1

Page 37 Exercise 1 Answer

Irrational numbers cannot be represented as a fraction.

Irrational numbers are non-terminating and non-repeating in nature.

Only terminating and repeating decimals can be represented as a fraction.

Rational numbers are terminating and repeating in nature.

Only rational numbers can be represented as a fraction while irrational numbers cannot.

Terminating and repeating decimals are known as rational numbers while non-repeating decimals are known as irrational numbers.

Envision Math Grade 8 Chapter 1 Topic 1 Solution Guide

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1 Page 37 Exercise 3 Answer

We need to find whether the number √8 is greater than, less than or equal to 4.

Finding where the given square root number lie in the number line, thus,

√4 < √8 < √9

2 < √8 < 3

Thus, the value of √8 is less than the number 3.

Therefore, the number √8 is less than the number 4.

The number √8 is less than the number 4.

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1

Page 37 Exercise 4 Answer

We need to solve m2 = 14

Solving the given expression, we get,

m2 = 14

m = √14

m = ±√14

The solutions are m = +√14 and m = −√14

Envision Math Grade 8 Chapter 1 Topic 1 Real Numbers Solutions

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1 Page 37 Exercise 6 Answer

We need to write \(1 . \overline{12}\) as a mixed number.

Writing the given numbers as a mixed fraction, we get,

x = 1.121212…

100x = 112.121212…

100x – x = 112.121212… – 1.121212…

99x = 112 – 1

99x = 111

\(x=\frac{111}{99}\) \(x=1 \frac{12}{99}\)

The number is represented as mixed fraction as \(x=1 \frac{12}{99}\)

Real Numbers Topic Solutions Grade 8 Envision Math

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1 Page 38 Exercise 1 Answer

Given that the table shows the results of the draw. The students who drew rational numbers will form the team called the Tigers. The students who drew irrational numbers will form the team called the Lions. We need to list the members of each team.

 

real numbers

 

The students who drew rational numbers will form the team called the Tigers.

The students who drew irrational numbers will form the team called the Lions.

Lydia drawn √38

The number 38 is not a perfect square. Hence, it is an irrational number.

Marcy drawn \(6.3 \overline{4}\)

The number is repeating in nature. Hence, it is a rational number.

Caleb drawn √36

The number 36 is a perfect square. Hence, it is a rational number.

Ryan drawn 6.343443444…

The number is non terminating and non-repeating. Hence, it is an irrational number.

Anya drawn \(6 . \overline{34}\)

The number is repeating in nature. Hence, it is a rational number.

Chan drawn √34

The number 34 is not a perfect square. Hence, it is an irrational number.

Lydia – Irrational number

Marcy – Rational number

Caleb – Rational number

Ryan – Irrational number

Anya – Rational number

Chan – Irrational number

Team Tigers:

Marcy, Caleb, Anya

Team Lions:

Lydia, Ryan, Chan

Team Tigers:

Marcy, Caleb, Anya

Team Lions:

Lydia, Ryan, Chan

Envision Math Grade 8 Chapter 1 Topic 1 Answers

Envision Math Grade 8 Volume Student Edition Solutions Chapter 1 Real Numbers Topic 1 Page 38 Exercise 2 Answer

Given that the student on each team who drew the greatest number will be the captain of that team. We need to find who will be the captain of the Tigers.

The numbers drawn by the members of the team Tigers will be,

Marcy – \(6.3 \overline{4}\)

Cleb – √36

Anya – \(6 . \overline{34}\)

Thus, the numbers are,

\(6.3 \overline{4}\) = 6.344444…

√36 = 6

\(6 . \overline{34}\) = 6.343434…

The greatest number among them will be \(6.3 \overline{4}\)

Therefore, Marcy will be the captain of the Tigers.

Marcy will be the captain of the Tigers.

 

Multiplying Rational Numbers: Solutions for Envision Math Grade 8 Exercise 1.5

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5

Envision Math Grade 8, Volume 1, Chapter 1: Real Number

Page 31 Exercise 1 Answer

Given that, Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model. We need to find the dimensions of the possible models that she can build. Also, find how many blocks would Janine use for each model.

Find the perfect cubes from 0 to 150

​1 × 1 × 1 = 1

2 × 2 × 2 = 8

3 × 3 × 3 = 27

4 × 4 × 4 = 64

5 × 5 × 5 = 125

6 × 6 × 6 = 216

The total number of blocks given is 150. Thus, all possible dimensions less than 150 are our solutions.

The dimensions of the possible models that she can build will be,

2 × 2 × 2

3 × 3 × 3

4 × 4 × 4

5 × 5 × 5

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 1 Real Number Exercise 1.5

Multiplying Rational Numbers: Solutions for Envision Math Grade 8 Exercise 1.5 Page 31 Exercise 2 Answer

We need to explain how the dimensions of a solid related to its volume.

The dimensions of the possible models that Janine can build will be,
​2 × 2 × 2

3 × 3 × 3

4 × 4 × 4

5 × 5 × 5

The dimensions of a solid are related to its volume since they are building a solid, cube-shaped model.

The volume of the cube will be V = s3 where s is the length of the edge of the cube.

The dimensions of a solid are related to its volume since the dimensions consist of nothing but the length of the edge of the cube.

 

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 31 Exercise 1 Answer

Given that, Janine wants to build a model using \(\frac{1}{2}\)-icnh cubes. We need to find how many \(\frac{1}{2}\)-icnh cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches.

Let x be the number of \(\frac{1}{2}\) inch cube with the side length of 4 inches.

The equation formed from the given information will be,

\(\frac{1}{2} x=4\)

x = 4 x 2

x = 8

Therefore, she needs 8 blocks per side.

Thus, the number of little cubes she needs will be,

V = 8 × 8 × 8 = 512

Therefore, she needs to use 512, \(\frac{1}{2} x=4\) inch cubes to build a cube-shaped model with a side length of 4 inches.

She needs to use 512, \(\frac{1}{2} x=4\) inch cubes to build a cube-shaped model with a side length of 4 inches.

 

Page 32 Question 1 Answer

We need to explain how we can solve equations with squares and cubes.

In order to solve equations with squares, find the number obtained by multiplying the given number two times itself.

For example, if the number is 4

The square of the given number is,

4 × 4 = 16

The square of 4 is 16

In order to solve equations with cubes, find the number obtained by multiplying the given number three times itself.

For example, if the number is 2

The cube of the given number is

2 × 2 × 2 = 8

The cube of 2 is 8

Solve equations with squares by finding the number obtained by multiplying the given number two times itself. Similarly, solve equations with cubes by finding the number obtained by multiplying the given number three times itself.

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 32 Exercise 1 Answer

We need to find the side length, s, of the square below.

The area of the square is A=100m2

square
Real Numbers Page 32 Exercise 1 Answer

 

The side of the square will be 10m.

 

Page 32 Exercise 2 Answer

We need to solve x3 = 64

Real Numbers Page 32 Exercise 2 Answer
The value of x = 4

 

Page 33 Exercise 3 Answer

We need to solve a3 = 11
Real Numbers Page 33 Exercise 3 Answer

 

The solutions are c=√27 and c=−√27

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 32 Exercise 1 Answer

We need to explain why are there two possible solutions to the equation s2 = 100 And also, we have to explain why only one of the solutions is valid in this situation.

Real Numbers Page 32 Exercise 1 Answer Image

Here, we have taken the value of s as +10 since we know that the value of the length cannot be a negative one.

The value of the length cannot be negative. This is why we have taken only the positive value of s

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 34 Exercise 2 Answer

Suri solved the equation x2 = 49 and found that x=7

We need to find the error Suri made.

Real Numbers Page 34 Exercise 4 Answer

The value of x = +7 and x = −7. The values are both positive and negative. This is the error Suri made.

We need to explain why are the solutions to x2 = 17 is irrational.
Real Numbers Page 34 Exercise 4 Answer Image

The solutions are irrational since the square root of 17 is not a perfect square number.

The solutions are irrational since the square root of 17 is not a perfect square.

 

Page 34 Exercise 5 Answer

Given that, If a cube has a volume of 27 cubic centimeters, We need to find the length of each edge by using the volume formula V = s3

Real Numbers Page 34 Exercise 5 Answer
The length of each edge will be 3cm.

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 34 Exercise 7 Answer

We need to solve x3 = −215

Real Numbers Page 34 Exercise 7 Answer
The solution is x = \(-\sqrt[3]{215}\)

 

Multiplying Rational Numbers: Solutions for Envision Math Grade 8 Exercise 1.5 Page 35 Exercise 9 Answer

We need to solve a3 = 216
Real Numbers Page 34 Exercise 9 Answer

The solution is a = 6

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 35 Exercise 12 Answer

We need to solve y2 = 81
Real Numbers Page 35 Exercise 12 Answer

The solutions are y = +9 and y = −9

 

Page 35 Exercise 13 Answer

We need to solve w3 = 1000

Real Numbers Page 35 Exercise 13 Answer

The solution is w = 10

 

Page 35 Exercise 14 Answer

The area of a square garden is given as A=121ft2. We need to find how long is each side of the garden.
Real Numbers Page 35 Exercise 14 Answer

The length of the side of the garden is 111ft

 

Page 35 Exercise 15 Answer

We need to solve b2 = 77
Real Numbers Page 35 Exercise 15 Answer

The solutions are b = +√77 and b = −√77

 

Page 36 Exercise 19 Answer

Given that, Manolo says that the solution of the equation g2 = 36 is g = 6 because 6 × 6 = 36

Check whether Manolo’s reasoning is complete or not
Real Numbers Page 36 Exercise 19 Answer

Therefore, Manolo’s reasoning is wrong.

 

Page 36 Exercise 20  Answer

We need to solve \(\sqrt[3]{-512}\)
Real Numbers Page 36 Exercise 20 Answer

The solution is \(\sqrt[9]{-512}=-8\)

We need to explain how we can check that our result \(\sqrt[3]{(-8)^3}\).

To verify the result, find the cube of the obtained result,

Thus, we get,

−8 × −8 × −8 = −512

Thus, our obtained result is correct.

The cube of -8 is -512. Hence, it is verified.

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 36 Exercise 21 Answer

Given that, Yael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%. We need to find the length, to the nearest tenth, of one side of the new garage.
Real Numbers Page 36 Exercise 21 Answer

The length, to the nearest tenth, of one side of the new garage is 18.5ft

 

Page 36 Exercise 22 Answer

Given that the Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot. We need to find how much will the hardwood floors cost.

The given volume is 6859ft3
Real Numbers Page 36 Exercise 22 Answer

The hardwood floor costs 3610 dollars.

 

 

Multiplying Rational Numbers: Solutions For Envision Math Grade 8 Exercise 1.5 Page 36 Exercise 23 Answer

Given that, while packing for their cross-country move, the Chen family uses a crate that has the shape of a cube.

If the crate has the volume V=64 cubic feet, we need to find the length of one edge.
Real Numbers Page 36 Exercise 23 Answer Image 1

The length of one edge will be 4ft

Given that, While packing for their cross-country move, the Chen family uses a crate that has the shape of a cube.

The Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet, we need to find the painting will fit flat againsts a side of the crate.
Real Numbers Page 36 Exercise 23 Answer Image 2

Thus, the area of the cube is more than that of the given square. Thus, it can fit flat on the floor.

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions

Page 19 Exercise 1 Answer

Given that, Courtney and Malik are buying a rug to fit in a 50-square-foot space. We need to determine which rug should they purchase.
Real Numbers Page 19 Exercise 1 Answer

We need to buy a rug to fit in a 50-square-foot space. Therefore, we need to buy a rug which is smaller than this.

The first rug will be,

7ft × 7ft = 49ft2

The second rug is of circle-shaped, thus, its area will be,

A = πr2

A = \(\frac{22}{7} \times 4^2\)

A = 50.2857

The third rug area will be,

⇒ \(6 \mathrm{ft} \times 8 \frac{1}{2} \mathrm{ft}=6 \mathrm{ft} \times \frac{17}{2} \mathrm{ft}\)

=3 × 17ft2

= 51ft2

​Only the first rug is smaller than the 50 square foot space. Therefore, it is

Courtney and Malik should purchase the first rug whose dimensions are 7ft x 7ft

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 1 Real Number Exercise 1.3

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 19 Exercise 1 Answer

We need to explain how we have decided which rug Courtney and Malik should purchase.

As we need to find a rug which has to be fit in the space of 50 square foot.

We need to buy a rug which is smaller than the space.

Thus, we need to find the areas of all three rugs and then decide which one’s area is less than 50 square foot.

The area of the first rug is, A=49ft2

The area of the second rug is, A=50.2857ft2

The area of the third rug is, A = 51ft2

We have decided which rug to buy by finding the areas of all three rugs and determining which one’s area is smaller than 50 square foot.

 

Page 20 Question 1 Answer

We need to explain how we can compare and order rational and irrational numbers.

We can compare and order rational and irrational numbers by the following ways:

Irrational numbers cannot be represented as a fraction.

Irrational numbers are non-terminating and non-repeating in nature.

Only terminating and repeating decimals can be represented as a fraction.

Rational numbers are terminating and repeating in nature.

Only rational numbers can be represented as a fraction while irrational numbers cannot.

Rational numbers are terminating and repeating while irrational numbers are non-terminating and non-repeating in nature.

Envision Math Grade 8 Exercise 1.3 Real Numbers Answers

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 20 Exercise 1 Answer

We need to determine which decimals we can use to find a better approximation.

Find the two closest consecutive perfect squares of the given number √74

The perfect squares which is closest to them is √64 and √81

Thus, the given number √74 is between the numbers 8 and 9

As we can see that the number √74 is closest to 9 more than the number 8.

Thus, ​8 < √74 < 9

8.5 < √74

Thus, squaring the decimals above 8.5 and below 9 we get,

8.6 × 8.6 = 73.96

8.7 × 8.7 = 75.69

8.8 × 8.8 = 77.44

8.9 × 8.9 = 79.21

Therefore, the number √74 is about 8.6ft

For finding the better approximation, square decimals between 8.5 and 8.9 since √74 > 8.5

 

Page 20 Exercise 1 Answer

We need to find between which two whole numbers is √12

The two closest perfect square roots of the given number √12 is,√9 and √16

Thus,

√9 < √12 < √16

3 < √12 < 4

Thus, the given number is between 3 and 4

The number √12 lies between the whole numbers 3 and 4

 

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 21 Exercise 2 Answer

We need to compare and order the following numbers:

√11, \(2 \frac{1}{4}\), −2.5, 3.6, −3.97621…

Finding where the given square roots numbers lie in the number line, thus,

√9 < √11 < √16

3 < √11 < 4

3 < √11 < 3.5

Since it is closer to the number 9 than 16.

\(2 \frac{1}{4}\) = 2.250

Thus, ordering the given numbers, we get,

\(-3.97621 \ldots<-2.5<2 \frac{1}{4}<\sqrt{11}<3 . \overline{6}\)

Comparing and ordering the given numbers, we get,

\(-3.97621 \ldots<-2.5<2 \frac{1}{4}<\sqrt{11}<3 . \overline{6}\)

 

Page 22 Exercise 1 Answer

We need to explain how we can compare and order rational and irrational numbers.

Irrational numbers cannot be represented as a fraction.

Irrational numbers are non-terminating and non-repeating in nature.

Only terminating and repeating decimals can be represented as a fraction.

Rational numbers are terminating and repeating in nature.

Only rational numbers can be represented as a fraction while irrational numbers cannot.

Find some rational number closer to the irrational number. Then compare the other one with the obtained estimate for the irrational number to order them.

Find some rational number closer to the irrational number. Then compare the other one with the obtained estimate for the irrational number to order them.

 

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 22 Exercise 3 Answer

We need to find which one is a better approximation of √20. The given approximated values are 4.5 or 4.47

Finding where the given square root number lie in the number line, thus,

√16 < √20 < √25

4 < √20 < 5

The number 20 is closer to 4 than 5.

Thus, 4 < √20 < 4.5

Thus, squaring the decimal values to find a better estimate, we get,

4.47 × 4.47 = 19.9809

4.5 × 4.5 = 20.25

Thus, 4.47 is a better estimate.

4.47 is a better estimate of √20

Real Number Solutions Grade 8 Exercise 1.3 Envision Math

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 22 Exercise 5 Answer

We need to approximate √18 to the nearest tenth and plot the number on a number line.

Finding where the given square root number lie in the number line, thus,

√16 < √18 < √25

4 < √18 < 5

4 < √18 < 4.5

since √18 is closer to 4 than 5.

Thus, squaring the decimal values to find a better estimate, we get,

4.1 × 4.1 = 16.81

4.2 × 4.2 = 17.64

4.3 × 4.3 = 18.49

4.4 × 4.4 = 19.36

Thus, the number 4.2 is the better estimate.
Real Numbers Page 22 Exercise 5 Answer
The number 4.2 is the better estimate.
Real Numbers Page 22 Exercise 5 Answer

 


Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 22 Exercise 6 Answer

We need to compare 5.7145… and √29 and find which one is greater than the other.

Finding where the given square root number lie in the number line, thus,

√25 < √29 < √36

5 < √29 < 6

5 < √29 < 5.5

The number √29 is closer to 5 than 6

Thus, squaring the decimal values to find a better estimate, we get,

5.2 × 5.2 = 27.04

5.3 × 5.3 = 28.09

5.4 × 5.4 = 29.16

Thus, 5.3 is a better estimate of √29

Therefore, √29 < 5.7145…

Comparing the numbers, we get, √29 < 5.7145…

 

Page 23 Exercise 9 Answer

We need to compare -1.9612… and -√5 and find which one is greater than the other.

Finding where the given square root number lie in the number line, thus,

−√4 < −√5 < −√9#

−2 < −√5 < −3

Therefore, ordering the given numbers we get,

−√5 < -1.96312…

Ordering the numbers, we get,

−√5 < -1.96312…

Envision Math Grade 8 Chapter 1 Exercise 1.3 Solutions

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 24 Exercise 13  Answer

Given that, Rosie is comparing √7 and 3.4444….. She says that

√7 > 3.4444… because √7 = 3.5.

We need to check whether the comparison is correct or not.

Finding where the given square root number lie in the number line, thus,

√4 < √7 < √9

2 < √7 < 3

2.5 < √7 < 3

Thus, √7 < 3.4444…

The comparison made by Rosie is incorrect.

Given that, Rosie is comparing √7 and 3.4444….. She says that

√7 > 3.4444… because √7 = 3.5.

We need to find the mistake Rosie likely made.

Finding where the given square root number lie in the number line, thus,

√4 < √7 < √9

2 < √7 < 3

2.5 < √7 < 3

Thus,

√7 < 3.444…

√7 = 2.646

The comparison made by Rosie is incorrect since the value of √7 = 2.646

 

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 24 Exercise 14 Answer

We need to approximate −√23 to the nearest tenth. Also, draw the point on the number line.

Finding where the given square root number lie in the number line, thus,

−√16 > −√23 > −√25

−4 > −√23 > −5

−4.5 > −√23 > −5

Thus, squaring the decimal values to find a better estimate, we get,

4.6 × 4.6 = 21.16

4.7 × 4.7 = 22.09

4.8 × 4.8 = 23.04

4.9 × 4.9 = 24.01

So, a better approximate is -4.7

The better approximate is -4.7
Real Numbers Page 24 Exercise 14 Answer

Envision Math 8th Grade Exercise 1.3 Step-By-Step Real Number Solutions

 

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 24 Exercise 15  Answer

Given that, the length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.

We need to find an irrational number represents the length of each side of the squares.

The given area of the rectangle is 90
square units.
Dividing the given area of the rectangle into two equal squares, we get,

\(\frac{90}{2}=45\)

The area of the square is 45 square units.

The length of each side of the square is,

Area = side2

45 = s2

s = √45

An irrational number represents the length of each side of the squares is √45

Given that the length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.

We need to estimate the length and width of the rectangle.

From part (a), we have divided the rectangle into two equal squares of length of each side is a = √45
Real Numbers Page 24 Exercise 15 Part (b) Answer

The width of the rectangle will be a

Thus,

a = √45

√36 < √45 < √49

6 < √45 < 7

6.5 < √45 < 7

Thus, squaring the decimal values to find a better estimate, we get,

6.6 × 6.6 = 43.56

6.7 × 6.7 = 44.89

6.8 × 6.8 = 46.24

Thus, a = 6.7

The length of the rectangle will be,

l = 2a

l = 2 × 6.7

l = 13.4

The width is l = a = 6.7

The length and width of the rectangle is 6.7,13.4 respectively.

How To Solve Real Numbers Exercise 1.3 In Envision Math Grade 8

Envision Math Grade 8 Volume 1 Chapter 1 Exercise 1.3 Real Number Solutions Page 24 Exercise 17  Answer

The area of a square poster is 31 square inches. We need to find the length of one side of the poster to the nearest whole inch.

The given area of the square poster is 31 square inches.

The length of one side of the poster is,

A = s2

31 = s2

s = √31

Finding where the given square root number lie in the number line, thus,

√25 < √31 < √36

5 < √31 < 6

The number √31 is closer to 6 than 5.

The length of one side of the poster to the nearest whole inch is 6.

The area of a square poster is 31 square inches. We need to find the length of one side of the poster to the nearest tenth of an inch.

The given area of the square poster is 31 square inches.

The length of one side of the poster is s=√31

Finding where the given square root number lie in the number line, thus,

√25 < √31 < √36

5 < √31 < 6

5.5 < √31 < 6

Thus, squaring the decimal values to find a better estimate, we get,

5.5 × 5.5 = 30.25

5.6 × 5.6 = 31.36

5.7 × 5.7 = 32.49

5.8 × 5.8 = 33.64

Therefore, the length of one side of the poster to the nearest tenth of an inch is 5.6

The length of one side of the poster to the nearest tenth of an inch is 5.6

 

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4

Envision Algebra 1 Assessment Readiness Workbook Chapter 4

Page 52 Exercise 1 Answer

In the question, five numbers are given as,

1. \(\frac{1}{7}\)

2. √3

3. √15

4. √25

5. \(\frac{2}{3}\)

It is required to find which of the given numbers are irrational.

To find this, consider each number and check whether it can be reduced to any ratio between an integer p and a natural number q. If it is possible, the number is not irrational.

Consider the number \(\frac{1}{7}\) and check whether it can be reduced to any ratio between an integer p and a natural number q.

The number \(\frac{1}{7}\) is in the form \(\frac{p}{q}\).

So, it is rational.

\(\frac{1}{7}\) is rational.

Consider the number √3 and check whether it can be reduced to any ratio between an integer p and a natural number q.

√3 cannot be reduced to any ratio between an integer p and a natural number q.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4

So, √3 is irrational.

Consider the number √15 and check whether it can be reduced to any ratio between an integer p and a natural number q.

√15 = √3 × √5, both terms are irrational since they cannot be reduced to any ratio between an integer p and a natural number q.

So,√15 is irrational.

Consider the number √25 and check whether it can be reduced to any ratio between an integer p and a natural number q.

√25 = 5

5 can be written in \(\frac{p}{q}\) form as \(\frac{5}{1}\).

So, √25 is rational.

Consider the number \(\frac{2}{3}\) and check whether it can be reduced to any ratio between an integer p and a natural number q.

The number \(\frac{2}{3}\) is in the form \(\frac{2}{3}\).

So, it is rational.

\(\frac{2}{3}\) is rational.

By considering each number and checking whether it can be reduced to any ratio between an integer p and a natural number q, it is found that numbers (B)√3 and  (C) √15 are irrational.

Envision Algebra 1 Chapter 4 Answer Key

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 52 Exercise 2 Answer

In the question, \(\frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}\) is given. And four forms are as

1. \(\frac{7^7}{3^6}\)

2. \(\frac{7^3}{3^6}\)

3. \(\frac{3^2}{7^3}\)

4. \(\frac{3^6}{7^7}\)

It is required to rewrite \(\frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}\) using positive exponents.

Given \(\frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}\)

Exponential formula \(a-b=\frac{1}{a^b}\)

Apply the exponential formula \(a-b=\frac{1}{a^6} \text { to } \frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}\)

⇒ \(\frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}=\frac{(7)^5}{(7)^2(3)^2(3)^4}\)

Apply the exponential formula \(\frac{a^m}{a^n}=a^{m-n} \text { and } a^m a^n=a^{m+n} \text { to } \frac{(7)^5}{(7)^5(3)^2(3)^4}\)

⇒ \(\begin{aligned}
\frac{(7)^5}{(7)^2(3)^2(3)^4} & =\frac{(7)^{5-2}}{(3)^{2+4}} \\
& =\frac{(7)^3}{(3)^6}
\end{aligned}\)

Using exponential equations \(\frac{1}{a^b}\), \(\frac{a^m}{a^n}\) = am-n and aman= am+n,

⇒ \(\frac{(3)^{-4}(7)^{-2}}{(3)^2(7)^{-5}}\) can be rewritten as \(\frac{(7)^{3}}{(3)^{6}}\)

Page 52 Exercise 3 Answer

In the question, an equation is given as ∣2x−3∣ − 4 = 3 and four solutions are given as

1. −3

2. −2

3. 3

4. 5

5. 6

It is required to find the solution of the given equation.

To find this, first find the modulus value of ∣2x−3∣. Then two values are obtained. Take these two values as two cases and find the value of x from both cases.

Given |2x-3|-4=3
|2x-3|-4=3
|2x-3|=3+4
|2x-3|=7

So, talking modulus 2x-3=±7

Take the value 2x-3 as 7 and find the value of x

2x-3=7
2x=7+3
2x=10

Divide 2 on both sides

⇒ \(\frac{2 x}{2}=\frac{10}{2}\)

Take the value 2x-3 as-7 find the value of x

2x-3=-7
2x=-7+3
2x=-4

Divide 2 on both sides

\(\frac{2 x}{2}=\frac{-4}{2}\)

x=-2

By using the equation ∣x∣ = ±x, the two solutions obtained are

(B) x = −2 and

(D)x = 5

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 52 Exercise 4 Answer

In the question, it is given that a line passes through two points (4,−1) and (−4,3).

Also, four equations are given as

1. y = \(\frac{1}{2} x-1\)

2. y = 2x – 1

3. y = \(\frac{-1}{2} x-1\)

4. y = -2x -1

It is required to find which equation represents a line perpendicular to a line that passes through (4,−1) and (−4,3).

To solve this, first find the slope of the line passes through (4,−1) and (−4,3) using the equation m = \(\frac{y_2-y_1}{x_2-x_1}\) and then use this slope to find the slope of the line perpendicular to the line passes through (4, -1) and (-4, 3) using the equation m1 = \(\frac{-1}{m}\).

The slope of the line passes through (4,-1) and (4,3) using the equation.

m= \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m=\frac{3-(-1)}{-4(-4)} \\
& m=\frac{3+1}{-4-(4)}
\end{aligned}\)

Reducing \(\frac{4}{-8}=\frac{-1}{2}\)

The slope of line passing through (4,-1) and (-4,3) is \(\frac{-1}{2}[\)

Slope of the line passing through (4,−1) and (−4,3) is m = \(\frac{-1}{2}\).

Find the slope m1 of the line perpendicular to the line having slope m using the equation m1= \(\frac{-1}{m}\).

m1 = \(-\frac{1}{\left(-\frac{1}{2}\right)}\)

= 2

Find the equation of the line with slope 2 from the given options.

Since the slope of the line is 2, the equation satisfying is y = 2x − 1.

The slope of the line passing through(4,−1) and (−4,3) using the equation m = \(\frac{y_2-y_1}{x_2-x_1}\) is found as m = \(\frac{-1}{2}\).

The slope of the line perpendicular to the line having slope m using the equation m1 = \(-\frac{1}{\left(-\frac{1}{2}\right)}\), is found as m1 = 2.

Since the slope of the line is 2, the equation satisfying is (B) y = 2x − 1.

Page 52 Exercise 6 Answer

In the question, an expression (6x4y6)2 is given. Also, four options are given as

1. 6x6y8

2. 12x8y12

3. 36x8y12

4. 36x6y8

It is required to simplify (6x4y6)2 and to find which of the four options is right.

To solve this, use the equation (abc)n = anbncn to the given expression and simplify it using mathematical operations.

Given (6x4y6

Apply the equation (abc)n = anbncn to (6x4y6

(6x4y6)² =(6)².(x4)².(y6
(6x4y6)²=36x8y12

So, (6x4y6)²=36x8y12

Using the exponential equations (abc)n = anbncn and

(ab)c= abc, (6x4y6)2 can be simplified as 36x8y12.

So (C) 36x8y12 is the correct option.

Page 53 Exercise 7 Answer

In the question, a function is given as f(x) = ∣−x+7∣.

Also, four piecewise functions are given as

1. g(x) = \(\left\{\begin{array}{l}
-x+7, \text { if } x \leq-7 \\
x-7 \text {,if } x>-7
\end{array}\right.\)

1. g(x) = \(\left\{\begin{array}{l}
-x+7 \text {, if } x \leq 7 \\
x-7, \text { if } x>7
\end{array}\right.\)

1. g(x) = \(\left\{\begin{array}{l}
-x+7 \text {, if } x \geq 0 \\
x-7 \text {, if } x<0
\end{array}\right.\)

1. g(x) = \(\left\{\begin{array}{l}
-x+7 \text {, if } x \geq 7 \\
x-7 \text {, if } x<7
\end{array}\right.\)

It is required to identify the piecewise function that has the same graph as the function f(x) = ∣−x+7∣.

To solve this, the function will break when the term in the modulus becomes zero.

Find the value of x when the modulus becomes 0 and find the function value to the left and right of the x value.

Find the value of x when the modulus becomes 0.

Given f(x)=|-x+7|

Find the value of when the models become o

=-x+7=0
x=7

so, the function breaks art x=7
find the function value when x<7.

​if x<7
f(x)=-(-x+7
f(x)=x-7
i.e., f(x)=x-7,if x<7

Find the function value when x ≥ 7.

f(x) = −x+7.

i.e., f(x) = −x+7,if x ≥ 7.

The piecewise function having the same graph as f(x) = ∣−x+7∣ is

1. g(x) = \(\left\{\begin{array}{l}
-x+7 \text {, if } x \geq 7 \\
x-7 \text {, if } x<7
\end{array}\right.\)

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 53 Exercise 8 Answer

In the question, a function f(x) = 2x is given.

It is required to state which type of function is f(x) = 2x

The general equation of the exponential function is f(x) = ax

Comparing the given function with the general equation of the exponential function f(x) = ax, it is found that

f(x) = 2x is an exponential function with the value of a 2.

By Comparing the given function with the general equation of the exponential function f(x) = ax, it is found that f(x) = 2x is an exponential function with the value of a is 2.

Page 53 Exercise 9 Answer

In question four scores of a student are given as 89, 98, 93, and 97 and the average of five scores is given as 94.

It is required to find what score must the student earn on the fifth test to have an average score of 94.

To find this, let the score of the fifth subject be x. Substitute this in the equation of average score and find x.

Let x be the fifth score and substitute it in the equation of the average score.

​Average score \(=\frac{\text { total score }}{\text { number of Scores }}\)

⇒ \(94=\frac{89+98+93+97+x}{5}\)

⇒ \(94=\frac{377+x}{5}\)

94×5=377+x

Simplify the above equation and find x

470=377+x
x=470-377
x=93

So, the fifth score is 93.

Letting the fifth score as x and substituting it in the equation of average score, the fifth score is found as 93 to get an average of 94.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 53 Exercise 10 Answer

In the question, a graph is given.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 53 Exercise 10 Answer Image 1

Four equations are given as

y = −∣3x∣ + 2

y = ∣3x+2∣

y = −3∣x+2∣

y = ∣3x∣ + 2

It is required to find the correct equation of the graph.

To find this, first find the coordinates from the given graph and find the equation of the line using the two-point form of a line (y – y1) = \(\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)\).

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 53 Exercise 10 Answer Image 2

Find the coordinates from the given graph.
The coordinates are (0,2), (\(\frac{2}{3}\), 0) and (\(\frac{-2}{3}\), 0).

Find the equation of the line passing through (0,2) and (\(\frac{2}{3}\), 0) using two-point form of the line.

⇒ \(\left(y-y_1\right)=\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)\)

⇒ \((y-2)=\left(\frac{0-2}{\frac{2}{3}-0}\right)(x-0)\)

⇒ \((y-2)=\frac{-2}{\frac{2}{3}} x\)

(y-2)=-3x

Find the equation of the line passing through (0,2)(0,2) and (\(\frac{-2}{3}\), 0) using the two-point form of the line.

⇒ \(\left(y-y_1\right)=\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)\)

⇒ \((y-2)=\left(\frac{0-2}{\frac{-2}{3}-0}\right)(x-0)\)

⇒ \((y-2)=\frac{-2}{\frac{-2}{3}} x\)

(y-2)=3x

Write the two equations in the modulus form.

The equations are (y−2) = −3x and

(y−2) = 3x

−3x ≤ y−2 ≤ 3x

So, the resultant equation can be written as y = ∣3x∣+2.

By finding the coordinates from the given graph and applying a two-point form of the line (y – y1) = \(\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)\), the equation of the graph is found as y = |3x|+2.

So, the correct equation is (D) y = |3x|+2.

Page 53 Exercise 11 Answer

In the question, a scatter point graph is given with four options.

1. y = x + 2

2. y = −x + 2

3. y = 2x + 2

4. y = −2x + 2

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 53 Exercise 11 Answer Image 1

 

It is required to find the best estimation of the equation of the fit line from given options.

To solve the question, draw a line that approximately divides the scatter points in half. Now find the equation of this line and compare it with the given options.

Draw a best-fit line of the scatter points.

The line should be drawn in this way so that it approximately divides the number of points in half on both sides of the line.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 53 Exercise 11 Answer Image 2

Find the equation of the line.

Extend the line so that it touches the x-axis. Now two points on the line are (0,2) and (−1,0).

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 53 Exercise 11 Answer Image 3

The slope of the line is

\(\frac{0-2}{-1-0}=\frac{-2}{-1}\)

= 2

And the y-intercept is 2.

Thus, the equation of the line is y = 2x + 2.

Therefore, option C) y = 2x + 2 is the best-estimated equation for the given scatter points which was solved using the general equation of line and slope of the line.

Envision Algebra 1 Assessment Readiness Workbook Chapter 4 Solutions

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 43 Exercise 12 Answer

In the question, six numbers \(\frac{-1}{4}\), -1, √3. √-8, \(\frac{1}{2}\), 3 are given.

It is required to arrange the numbers from least to greatest.

To solve the question, convert all numbers in decimal format. Then arrange the number from least to greatest.

Arrange the numbers from least to greatest.

Numbers are \(\frac{-1}{4}\), -1, √3, √-8, \(\frac{1}{2}\), 3.

Using a calculator, write numbers as decimals, -0.25, -1, 1.732, -2.828, 0.5, 3.

Now arrange in ascending order −2.828, −1, −0.25, 0.5, 1.732,3

Thus, the numbers order from least to greatest are −√8, -1, \(\frac{-1}{4}\), \(\frac{1}{2}\), √3, 3.

Thus, the numbers order from least to greatest are −√8, -1, \(\frac{-1}{4}\), \(\frac{1}{2}\), √3, 3 which was solved by first converting numbers into decimal and then arranging.

Page 54 Exercise 13 Answer

In the question, a dot plot of data is given.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 13 Answer Image 1

It is required to find the mean and median of the data shown in the above figure to the nearest hundredth.

To solve the question, make a table of given data. Then find the mean and median using suitable formulas and mathematical operations.

Make a table of the given data.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 13 Answer Image 2

Find the mean of the data.

Mean \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\)

⇒ \(\bar{x}=\frac{138}{13}\)

⇒\(\bar{x}=11.5\)

Thus, the mean of the data nearest to the hundredth is 11.50.

Find the median of the data.

Since the sum of the frequencies n is 13, which is an odd number.

Then the median is at the position

⇒ \(\left(\frac{13+1}{2}\right)^{t h}=7^{t h}\).

The number in the 7th position is 10.

Therefore, the median nearest to the hundredth is 10.00.

Therefore, the mean of the data nearest to the hundredth is $11.50$ and the median nearest to the hundredth is $10.00$. They are different as the mean gives the average value of data and the median gives the middle number of data which were solved using formulas of mean and median.

Page 54 Exercise 14 Answer

In the question, the area of the rectangle is represented by a quadratic equation, 12x2 + x − 6, and the width of the rectangle is represented as 3x−2.

It is required to find the width of the rectangle.

To solve the question, find the roots of the quadratic equations using quadratic formulas. Then write the equation as a product of its factors using its roots.

Find the roots of the quadratic equation.

Given equation is 12x²-x+6

⇒ \(x=\frac{-1 \pm \sqrt{1^2-4(12)(6)}}{2(12)}\)

⇒ \(\begin{aligned}
& x=\frac{-1 \pm \sqrt{1+288}}{24} \\
& x=-1 \pm 17
\end{aligned}\)

Solving further to calculate roots

⇒ \(\begin{aligned}
& x=\frac{-1+17}{24} \\
& x=\frac{16}{24} \\
& x=\frac{2}{3}
\end{aligned}\)

Solving further to calculate another root

⇒ \(\begin{aligned}
& x=\frac{-1-17}{24} \\
& x=\frac{-18}{24} \\
& x=\frac{-3}{4}
\end{aligned}\)

Solve further take x= \(\frac{2}{3}\)

Multiply by 3 and then subtract by 2

3x-2=0

⇒ \(x=\frac{-3}{4}\)

Multiply by 4 and then add by 3 4x+3=0.

Find the length of the square.

Now, the quadratic equation can be factored as

12x2 + x − 6 = (4x+3)(3x−2)

On comparing with the formula,

A = l × b

So, the length is 4x+3 units.

Thus, the length of the rectangle is 4x+3 units which was solved using the quadratic formula and then finding factors of the equation.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 54 Exercise 15 Answer

In the question, a linear inequality, 2x+4y < 4, is given.

It is required to graph the inequality.

To solve the question, rearrange the equation so that on the left side there is a y variable, and everything else on the right. Then plot the inequality as a simple linear equation. Then shade the graph above the line for a ‘greater than’ inequality and below the line for a ‘less than’ inequality.

Rearrange the equation.

It should be done so that on the left side there is a y variable, and everything else on the right.

Given inequality is 2x+4y<4

Subtract 2x from both sides

4y<4-2x

Divide by 4 on both sides

⇒ \(\begin{aligned}
\frac{4 y}{4} & <\frac{4}{4}-\frac{2 x}{4} \\
y & <1-\frac{1}{2} x
\end{aligned}\)

A simple linear equation of the above inequality can be written as y = 1 – \(\frac{1}{2} x\).

Plot the equation on the graph.

y = 1 – \(\frac{1}{2} x\)

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 15 Answer Image 1
Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 15 Answer Image 2
The line in the above graph is dashed as the inequality 4y < 4−2x contains < not ≤.

Solve further.

As the inequality, 4y < 4 − 2x, contains less than sign, the shaded area is

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 15 Answer Image 3

Thus, the graph of the linear inequality 2x+4y < 4 is sketched which was solved by first plotting a simple equation and then shading the region.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 15 Answer Image 3

 

Page 54 Exercise 16 Answer

In the question, a function f(x) = x2 − 4x + 5 is given.

It is required to graph the function.

To solve the question, convert the given equation into a general equation of the parabola.

Convert the given equation into a general equation of the parabola.

Given function can be written as

y = x2 − 4x + 5

Now add and subtract the square of half of the coefficient of x in the equation.

Given function f(x) = x²-4x+5
Given function can be written as

y=x2-4x+5+22-22

y=x2-4x+4+5-4
y=x2-4x+4+5-4
y=x2-4x+4+1

Solve further to get it in general form

y= x2-2(2) x+22+1
y=(x-2)2+1

Find the value of the vertex of the parabola.

y = (x−2)2 + 1

By comparing it with the standard equation, the vertex (h,k) of the parabola is (2,1).

As a > 1, then the graph opens up.

Now plot the graph.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 16 Answer Image 1

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 16 Answer Image 2

Therefore, a graph of the function f(x) = x2 − 4x + 5 is sketched which was solved using the standard equation of parabola.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 54 Exercise 16 Answer Image 2

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 55 Exercise 17 Answer

In the question, it is given that a ball is thrown directly upwards from a height of 2 feet and velocity is 16 feet per second, and equation h = −16t2 + 16t + 2.

It is required to find the time it takes for a ball to reach its maximum height and maximum height of the ball.

To solve the question, find the slope of the given equation and equate the obtained equation to zero to find the time t. Then substitute the obtained value of time in the given equation to find the maximum height.

Find the time that the ball takes to reach maximum height.

Given equation is h=-16t2+16t+2

⇒ \(\frac{d h}{d t}=-32 t+16\)

At maximum height \(\frac{d h}{d t}=0\)

-32t+16=0

Solve further
Subtract 16 from both sides
-32t=-16

Solve further
Subtract 16 from both sides

-32t=-16

Divide both sides by -32

⇒ \(\begin{gathered}
\frac{-32 t}{-32}=\frac{-16}{-32} \\
t=\frac{1}{2}
\end{gathered}\)

Thus, the time required to go to maximum height is 0.5 seconds.

Put t=0.5 in the equation

h=-16t2+16t+2
h=-16(0.5)2+16(0.5)+2
h=-16(0.25)+8+2
h=-4+10

Solve further

h=-4+10
h=6

Thus, the time required to go to maximum height is 0.5 second

The initial height is 2 feet.

So, the maximum height is

6 + 2 = 8 feet.

A. The ball takes 0.5 sec to reach its maximum height.

B. Maximum height reached is 8 feet which was solved using the concept of slope and first derivative.

Page 55 Exercise 18 Answer

In the question, a table is given.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 55 Exercise 18 Answer Image 1

It is required to plot the data and find the estimation of the equation of the best-fit line.

To solve the question, plot the scatter points on a graph. Then draw a line that approximately divides the scatter points in half. Now find the equation of this line.

Plot the scatter points on a graph using a given table.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 55 Exercise 18 Answer Image 2

 

Draw a best-fit line of the scatter points.

The line should be drawn in this way so that it approximately divides the number of points in half on both sides of the line.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 55 Exercise 18 Answer Image 3

Find the equation of the line.

Extend the line so that it touches the x-axis. Now two points on the line are (0,6) and (6,0).

Slope of the line is

\(\frac{0-6}{6-0}=\frac{-6}{6}\)

= -1

And the y-intercept is 6.

Thus, the equation of the line is y = −x + 6.

Therefore, the graph of scatter points is plotted and y = -x + 6 is the best-estimated equation for the given scatter points which was solved using a general equation of the line and slope of the line.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 55 Exercise 18 Answer Image 4

Envision Algebra 1 Student Edition Chapter 4 Practice Problems

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 56 Exercise 19 Answer

In the question,  it is given that the length of the hypotenuse of the right triangle is 25cm and the measure of one leg is 20 cm.

It is required to find the length of the other leg.

To solve the question, put the value of the lengths of the leg and hypotenuse in the formula of Pythagoras’ theorem. Then solve using appropriate mathematical operations.

Find the length of the leg of the right triangle.

Put b as 20 and c as 25 in the Pythagoras theorem formula.

​a²+b²=c²
put b=20 and c=25

a²+20²=25²
a²+400=625

Subtract 400 from both sides

a²+400-400=625-400

a²=225

Take square root on both sides

⇒ \(\begin{aligned}
\sqrt{a^2} & =\sqrt{225} \\
a & =\sqrt{25^2} \\
a & = \pm 15
\end{aligned}\)

But measuring length cannot be negative. Thus, the measure of another leg of the right triangle is 15 units.

Thus, the measure of another leg of the right triangle is 15 units which was solved using Pythagoras theorem.

Page 56 Exercise 20 Answer

In the question, a quadratic equation 5x2 − 18x + 9 = −3 is given.

1. 0.37

2. 0.6

3. 0.88

4. 2.72

5. 3

6. 3.23

It is required to choose the correct solutions of the equation from the given options.

To solve the question, convert the given quadratic equation into the form ax2 + bx + c = 0. Then put values of a, b, and c in the quadratic formula to find the solutions.

Convert quadratic equations into general form.

Given 5×2-18x+9=-3
Add 3 on both sides
5×2-18x+9+3=-3+3
5×2-18x+12=0

Compared with the general equation

a=5,b=-18,c=12

⇒ \(x=\frac{-(-18) \pm \sqrt{(-18)^2-4(5)(12)}}{2(5)}\)

⇒ \(\begin{aligned}
& x=\frac{18 \pm \sqrt{324-240}}{10} \\
& x=\frac{18 \pm \sqrt{84}}{10}
\end{aligned}\)

use to calculate to evaluate

⇒ \(x=\frac{18 \pm 9.16}{10}\)

Solve further

⇒ \(\begin{aligned}
& x=\frac{18+9.16}{10} \\
& x=2.716 \\
& x=\frac{18-9 \cdot 16}{10} \\
& x=0.884
\end{aligned}\)

Thus, the solutions of the equation to the nearest hundredth are 2.72 and 0.88.

Thus, solutions of the equation to the nearest hundredth are C) 2.72 and D) 0.88 which were solved using a quadratic formula.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 56 Exercise 21 Answer

In the question it is given two equations namely, y = 3x + 3 and y = 3x.

It is required to conclude that how the graph of the equation y = 3x + 3 can be described as the transformation of the graph of equation y = 3x. To solve the question start with writing the given equations as f(x) and g(x) respectively.

Then, write the value of g(x) in terms of f(x) and compare it with the formula of vertical shift to get the required answer.

Write the first equation as f(x) and g(x) as the second equation.

​f(x) = 3x + 3

g(x) = 3x

Write f(x) in terms of g(x).

​f(x) = 3x + 3

f(x) = g(x) + 3

Now, rewrite the expression and find the value of g(x) in terms of f(x) by subtracting 3 from both sides of the equation.

​g(x) + 3 − 3 = f(x) − 3

g(x) = f(x) − 3

Use the formula that vertical shift depends upon the value of k.

g(x) = f(x) + k, which means the graph is shifted k units upwards.

g(x) = f(x) − k, which means the graph is shifted k units downwards.

From the above-calculated equation, it is clear that for the given equations g(x) = f(x) − 3, the shift is 3 units downwards.

The transformation of graph y = 3x + 3 as y = 3x can be described by a 3-unit downwards translation, so the correct option is C) translation 3 units down which was solved using the formula for the vertical shift.

Page 56 Exercise 22 Answer

In the question, it is given an equality as −2x + 5 < 1.

It is required to graph the solution of the inequality.

To solve the question, start with writing the given inequality and solve it using basic mathematical operations.

Finally, plot the obtained answer on the graph.

Write the given inequality.

Given inequality -2x+5<1

Add 2x to both sides

-2x+2x+5<1+2x

5<1+2x

Subtract 1 from both sides

5-1<1+2x-1

4<2x

Divide 2 on both sides

⇒ \(\frac{4}{2}<\frac{2 x}{2}\)

2<x

So the solution of the inequality lies on the right-hand side of the value 2 on the graph.

Plot the graph.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 56 Exercise 22 Answer

The graph for the given inequality i.e. −2x + 5 < 1 is as shown and the solution for the equation is given by x > 2.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 56 Exercise 22 Answer

Page 56 Exercise 24 Answer

In the question it is given that $500 has been deposited in an account that gives 2% interest compounded monthly.

It is required to use the formula A = \(P\left(1+\frac{r}{n}\right)^{n t}\) to determine how much money will be in the account after 5 years. To solve the question start with writing the given information in the question. Then substitute the given values in the formula. Simplify the formed equation using basic mathematical operations and a calculator to get the required answer.

According to the given

p=500
r=2%
n=12
t=5 years

Now substitute the given values into the formula

A= \(P\left(1+\frac{r}{n}\right)^{n t}\)

A= \(500\left(1+\frac{2}{12}\right)^{12(5)}\)

A= \(500\left(1+\frac{1}{6}\right)^{60}\)

A= \(500\left(\frac{6+1}{6}\right)^{60}\)

A=\(500\left(\frac{7}{6}\right)^{60}\)

A+500(10394.59)
A=51,97,295

So the amount calculated is $51,97,295.

When $500 is deposited as a principle in a bank with 2% interest compounded monthly, then after 5 years the amount accumulated in the account is calculated to be $51,97,295 which was solved using the formula A = \(P\left(1+\frac{r}{n}\right)^{n t}\)

Page 57 Exercise 25 Answer

In the question it is given two equations namely, y = \(\sqrt[3]{\frac{x}{6}}\) and y = \(\sqrt[3]{x}\).

It is required to conclude how the graph of the equation y = \(\sqrt[3]{\frac{x}{6}}\) can be described as the transformation of the graph of the parent function y = \(\sqrt[3]{x}\).

To solve the question start with writing the given equations as f(x) and g(x) respectively. Then, write the value of g(x) in terms of f(x) and compare it with the formula of the horizontal shift to get the required answer.

Write the parent equation as f(x) and g(x) as the second equation.

f(x) = \(\sqrt[3]{x}\)

g(x) = \(\sqrt[3]{\frac{x}{6}}\)

Now, rewrite the expression and find the value of g(x) interms of f(x) by substituting the value of x.

g(x) = \(\sqrt[3]{\frac{x}{6}}\)

g(x) = \(\sqrt[3]{\frac{1 x}{6}}\)

g(x) = \(f\left(\frac{x}{6}\right)\)

g(x) = \(f\left(\frac{1}{6} x\right)\)

Use the formula that for a parent function f(x). A new function g(x) = a⋅f(x), where a is a constant, results in a horizontal compression or vertical stretch of the function f(x).

a > 1, means the graph is horizontally stretched.

0 < a < 1, means the graph is horizontally compressed.

From the above-calculated equation, it is clear that for the given equations g(x) = \(f\left(\frac{1}{6} x\right)\).

so, a = \(\frac{4}{2}\)

Clearly. 0 < a < 1, so there is a horizontal compression by a factor of \(\frac{1}{a}\).

\(\frac{1}{a}\) = \(\frac{1}{\frac{1}{6}}\)

\(\frac{1}{a}\) = 6

Hence for the given equations of two graphs, the transformation from the parent function is horizontal compression by a factor of 6.

The transformation of graph \(\sqrt[3]{x}\) as y = \(\sqrt[3]{\frac{x}{6}}\) can be described by a horizontal compression by a factor of 6, so the correct option is B) horizontal compression by a factor of 6, which was solved using the formula for horizontal compression and stretch.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 57 Exercise 26 Answer

In the question, it is given a system of equations as y = −2x − 4 and y = −4x − 2.

It is required to find the solution of the system of these equations.

To solve the question, start by writing the given system of equations and substituting the value of y from one equation to another. Then, simplify the expression using basic mathematical operations until the value of x is obtained. Then substitute the value of x in any of the given equations to get the required answer.

​Given a system of equations

y=-2x-4
y=-4x-2
-4x-2=-2x-4

Add 4x on both sides

-4x-2+4x=-2x+4x-4
2x=2

divide 2 on both sides

⇒ \(\frac{2 x}{x}=\frac{2}{x}\)

x=1

Substitute this calculated values of x

y=-2(1)-4
y=-2-4
y=-6

So, the solution for the given system of equations is (x,y) = (1,−6).

The given system of equations i.e. y = −2x − 4 and y = −4x − 2 results in a solution value of x as 1 and y as −6 which was solved using basic mathematical operations.

Page 57 Exercise 27 Answer

In the question, a graph is given as

Also, four equations are given as

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 57 Exercise 27 Answer

a. y = \(\sqrt{x+5}\) − 3

b. y = \(\sqrt{x-3}\) + 5

c. y = \(\sqrt{x-5}\) + 3

d. y = \(\sqrt{x+3}\) + 5

It is required to state which of the given options is an equation for the transformation of the graph of the parent function y = √x shown in the given graph.

To solve this, find the origin of the given graph and find the equation of the curve using the shifted origin.

Find the origin of the curve from the given graph and find the equation of the curve using the shifted origin.

The given graph is

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 57 Exercise 27 Answer

From the graph origin of the curve is (−5,−3).

So, the equation of the curve is

⇒ \(\begin{aligned}
& y-(-3)=\sqrt{x-(-5)} \\
& y+3=\sqrt{x+5} \\
& y-(-3)=\sqrt{x-(-5)}
\end{aligned}\)

⇒ \(\begin{aligned}
& y+3=\sqrt{x+5} \\
& y=\sqrt{x+5}-3
\end{aligned}\)

So, option (A) is the correct option.

Option (A) y = \(\sqrt{x+5}\) – 3 is an equation for the transformation of the graph of the parent function y = √x shown in the given graph.

Page 57 Exercise 28 Answer

In the question, it is given that the formula for the volume of a cylinder in cubic feet is V = πr2h where r is the radius of the base in feet and h is the height in feet. Also, it is given that its volume is 1078πft3 and its height is 22 feet.

It is required to find the diameter of the cylinder.

To solve this, substitute the given volume and height in the equation V = πr2h and find r.

Then using this r, find the diameter d = 2r.

v=πr2h-1
v=1078π
h=22

Substitute the value in 1

v=πr2h
1078π=π.r2.22
Divide 22 on both sides

⇒ \(\begin{aligned}
& \frac{22 r^2}{22}=\frac{1078}{22} \\
& r^2=49
\end{aligned}\)

Square root on both sides

⇒ \(\begin{aligned}
\sqrt{r^2} & =\sqrt{49} \\
r & = \pm \sqrt{7^2} \\
r & = \pm 7
\end{aligned}\)

Since the radius is a length, it is positive

So,r=7

Diameter d=2r
d=2(7)
d=14

So, the diameter is 14ft.

By substituting the volume and height in the equation of volume, the radius of the cylinder is found as r = 7, and using this radius diameter is found as 14ft.

How To Solve Envision Algebra 1 Chapter 4 Questions

Page 57 Exercise 30 Answer

In the question, four pairs of points and four slopes are given as follows

(3, 1) and (5, -7) \(\frac{1}{4}\)

(10, 14) and (-2, 11) 4

(-11, 9) and (-12, 5) \(\frac{-1}{4}\)

(-2, 7) and (-10, 9) -4

It is required to match each pair of points to the slopes.

To match this, find the slope of each line using the formula m = \(\frac{y_2-y_1}{x_2-x_1}\) and match them according to the calculated slope.

Find the slope m1 of the line passing through (3,1) and (5,7)

⇒ \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m_1=\frac{-7-1}{5-3} \\
& m_1=\frac{-8}{2} \Rightarrow m_1=-4
\end{aligned}\)

Find the slope m2 of the line passing through (10,14) and (-2,11)

⇒ \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m=\frac{11-14}{-2-10} \\
& m_2=\frac{-3}{-12} \Rightarrow m_2=\frac{1}{4}
\end{aligned}\)

Find the slope m3 of the line passing through (-11,9) and (-12,5)

⇒  \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m_3=\frac{5-9}{-12+11} \\
& m_3=\frac{-4}{-1} \Rightarrow m_3=4
\end{aligned}\)

Find the slope of the line passing through (-2,7) and (-10,9)

⇒ \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m_4=\frac{9-7}{-10+2} \\
& m_4=\frac{2}{-8} \\
& m_4=\frac{-1}{4}
\end{aligned}\)

Match the points and the calculated slopes.

(3, 1) and (5, -7) -4

(10, 14) and (-2, 11) \(\frac{1}{4}\)

(-11, 9) and (-12, 5) 4

(-2, 7) and (-10, 9) \(\frac{-1}{4}\)

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 58 Exercise 31 Answer

In the question, it is given that the ages of the members of a beach clean-up club are 16, 19, 22, 25, 31, 33, 38 and 42.

Also, it is given that a new member who is $19$ years old joins the club.

It is required to describe how the age of the new member will affect the mean, median, mode, and range of the ages of the club.

To solve this, first find the mean, median, mode, and range of the ages without the age of the new member. Then find the mean, median, mode, and range of the ages including the age of the new member. Compare the obtained answers in both cases to describe how the age of the new member will affect the mean, median, mode, and range.

Find the mean of the ages without the age of the new member.

Mean = \(\frac{16+19+22+25+31+33+38+42}{8}\)

Mean= \(\frac{226}{8}\)

Mean= 28.25

Find the median ages without age of new member

The middle numbers are 25 and 31

Median= \(\frac{25+31}{2}\)

Median= \(\frac{56}{2}\)

Median=28

Find the mode of the ages without the age of the new member.

Mode is the highest occurring number. Here every number occurs only once.

So, no mode exists.

Find the range of the ages without the age of the new member.

​Range = 42 − 16

= 26

Find the mean of the ages with the age of the new member.

mean = \(\frac{16+19+19+22+25+31+33+38+42}{9}\)

Mean= \(\frac{245}{9}\)

Mean= 27.22

Find the median of the ages with the age of the new member.

The middle number is 25.

Median = 25

Find the mode of the ages with the age of the new member.

​Mode = highest occurring number

= 19

Find the range of the ages with the age of the new member.

​Range = 42 − 16

= 26

The addition of new members with 19 years old will reduce the mean as 19 is small relative to the given data.

Median becomes a single number as there are an odd number of terms.

Mode is possible when the new member is added as there are two numbers of the same value.

The range will be the same as the highest and lowest numbers are the same.

Page 58 Exercise 32 Answer

In the question, an expression 8x5 − 26x4 + 6x3 is given.

It is required to find the factored form of 8x5 − 26x4 + 6x3 and to explain it.

To solve this, first take 2x3 common from the given expression. Then factorize the remaining function.

Given expression 8x5 − 26x4 + 6x3

Take 2x3 common from the expression

8×5-263=2×3(4×2-13x+3)

Factorise 4×2-13x+3

4×2-13x+3=4×2-12x-x+3
=4x(x-3)-1(x-3)

Take(x-3) common from the above equation
4x(x-3)-1(x-3)=(4x-1)(x-3)

take (x-3) common from the above equation

By taking 2x3 common and factorizing the remaining term, the factorization of 8x5 − 26x4 + 6x3 is found as 2x3(x−3)(4x−3).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 58 Exercise 33 Answer

In the question, it is given that the measure of angle A of an obtuse triangle is two times the measure of angle B, while the measure of angle C is 28° less than the measure of angle B.

It is required to explain what are the degree measures of the three angles.

To find this, take ∠A as 2∠B and ∠C as ∠B−28 in ∠A + ∠B + ∠C = 180° and find ∠B. Using the value of ∠B find the other two angles.

∠A+∠B+∠C=180°

Substitute ∠A=2∠B
∠C=∠B-28°

2∠B+∠B+∠B-28°=180°
4∠B=180°+28°
4∠B=208°

Divide 4 on both sides

⇒ \(\frac{4 \angle B}{4}=\frac{208^{\circ}}{4}\)

∠B=52°

Find ∠A by substituting the value of ∠B as 52°

∠A=2∠B
∠A=2(52)°
∠A=104°

Find ∠C by substituting the value of ∠B is 52°

∠C=∠B-28°
∠C=52°-28°
∠C= 24°

By taking ∠A as 2∠B and ∠C as ∠B − 28 in ∠A+∠B+∠C = 180°, the values of the three angles are found as

∠A=104°

∠B = 52°

∠C = 24°

Page 58 Exercise 34 Answer

In the question, a function f(x) = 9x2 + 15x − 6 is given.

It is required to find what are the zeros of the function f(x) = 9x2 + 15x − 6.

To find the zeros, put f(x) = 0 and solve using mathematical operations.

Put 9x2 + 5x − 6 = 0 and simplify to find the zeros.

Given function f(x)=9x2+15x-6

⇒ \(\begin{aligned}
& 9 x^2+15 x-6=0 \\
& 9 x^2+18 x-3 x-6=0 \\
& 9 x(x+2)-3(x+2)=0 \\
& (9 x-3)(x+2)=0
\end{aligned}\)

​Take 3 common from (9x-3)

(9x-3)(x+2)=0
3(3x-1)(x2)=0
(3x-1(x+2)=0

Here(3x-1) is o or (x+2) is o by the  property, if ab=0

Put(3x-1)=0 and find x

3x-1=0
3x=1
⇒ \(x=\frac{1}{3}\)

Put (x+2)=0 and find x
x+2=0
x=-2

By taking 9x2 + 15x − 6 = 0, the zeros of the function are found as x = \(\frac{1}{3}\) and x = −2.

Page 59 Exercise 35 Answer

In the question, it is given that Dakota has a combination of 18 dimes and quarters for a total of $2.70.

It is required to write a system of equations used to determine the number of each type of coin Dakota has and to determine how many coins of each type Dakota has.

To solve this, let the number of dimes as x and the number of quarters as y. The given condition is Dakota has a combination of 18 dimes and quarters. Convert the condition in terms of $. Then solve the two equations to find x.

Convert the given condition Dakota has a combination of $18$ dimes and quarters to equation form.

x+y=18
Put 1dime=$0.1
1quarter=$0.25 given condition

x+y=18
0.1+0.25y=2.7
x+2.5y=27

Solve the two equations x+y=18 and

x+2.5y=27
x.x+2.5y-y=27-18
1.5y=9
y=6

Substitute the value of y as 6

x+y=18
x+6=18
x=18-6
x=12

The system of equations used to determine the number of each type of coin Dakota has been x + y = 18.

The number of dimes is 12 and the number of quarters is 6.

Page 59 Exercise 36 Answer

In the question, a piecewise-defined function is given as

f(x) = \(\left\{\begin{array}{l}
4 x+7, \text { if } x<-2 \\
-x, \text { if }-2<x \leq 2 \\
1, \text { if } x>2
\end{array}\right.\)

It is required to graph the given piecewise function.

To plot this first take y = f(x) and find the coordinates by substituting different numbers in the range to the specified value of the function.

​Put x as -3 in the function y=4x+7

y=4(-3)+7
y=-12+7
y=-5

So, the coordinates are (-3,-5)

Put x as -4 in the function y=4x+7

y=4(-4)+7
y=-16+7
y=-9

So, the coordinates are (-4,9)

Put x as -1 in the function y=-x
y=-x
y=-(-1)
y=1

So, the coordinates are (-1,1)

Put x as 2 in the function y=x
y=-x
y=-2
so, the coordinates is (2,-2)

Put x as 3 in the function y = 1.

The coordinate is (3,1).

Put x as 4 in the function y = 1.

The coordinate is (4,1).

Plot the calculated point in the coordinate plane and draw the graph.

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 59 Exercise 36 Answer

The points are plotted on the coordinate plane as

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 59 Exercise 36 Answer

In the question, a function

f(x) = \(\left\{\begin{array}{l}
4 x+7, \text { if } x<-2 \\
-x, \text { if }-2<x \leq 2 \\
1, \text { if } x>2
\end{array}\right.\) is given.

It is required to find the domain and range of the function.

To solve the question, use the fact that the domain of all the polynomials is (−∞,∞).

Find the range for each interval separately and then take their union.

Find the domain of the function.

As it is given that

f(x) = \(\left\{\begin{array}{l}
4 x+7, \text { if } x<-2 \\
-x, \text { if }-2<x \leq 2 \\
1, \text { if } x>2
\end{array}\right.\)

Now, the function f(x) has a polynomial at every interval. So, the domain of the function is (−∞,∞).

Find the range of the function.

For x < −2, f(x) = 4x + 7

Also, ​

​f(−2) = 4(−2) + 7

= −1

Thus the range of the function is (−∞,−1) when x < −2.

For −2 < x ≤ 2, f(x) = −x

Thus the range of the function is [−2,2] for −2 < x ≤ 2.

Solve further to get the range of function f(x).

For x > 2, f(x) = 1.

So, the range of the function for the interval x > 2 is 1.

Now, the range of the function f(x),

​R = (−∞,−1)∪[−2,2)∪{1}

= (−∞,2)

Thus, the range of the function f(x) is (−∞,2).

Therefore, the domain of the function f(x) is (−∞,∞) and the range is (−∞,2) which were solved according to definitions of domain and range.

In the question, a function

f(x) = \(\left\{\begin{array}{l}
4 x+7, \text { if } x<-2 \\
-x, \text { if }-2<x \leq 2 \\
1, \text { if } x>2
\end{array}\right.\) is given.

It is required to find out on which interval function is increasing or decreasing.

To solve the question, for each interval find the first derivative of the function. If f′(x) < 0, then the f(x) is increasing on that interval. And if f′(x) > 0, then the f(x) decreases on that interval.

Find the intervals where function is increasing or decreasing.

For x < −2, f(x) = 4x + 7

​f′(x) = 4

f′(x) > 0

So, f(x) is decreasing on the interval (−∞,−2).

Solve further for more intervals.

For −2 < x ≤ 2, f(x) = −x

​f′(x) = −1

f′(x) < 0

So, f(x) is increasing on the interval [−2,2].

Solve further for more intervals.

For x > 2, f(x) = 1.

f′(x) = 0

Since f′(x) = 0 the function is constant in the interval (2,∞).

Therefore, function f(x) is decreasing on the interval (−∞,−2) and increasing on the interval [−2,2] which were solved by calculating the first derivative of the function at each interval.

Envision Algebra 1 Chapter 4 Step-By-Step Solutions

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 4 Page 60 Exercise 1 Answer

In the question, it is given that a person named Kayden is considering three different ways to rent movies and video games.

a. Rent movies and game disks for $2 each.

b. Unlimited games and movies to any device for $20 per month.

c. Rent movies and game disks for a $10 monthly fee plus $2.5 per disk.

It is required to write the functions that give the cost to rent x movies or disks per month for each plan, that is A(x), B(x), and C(x). Then graph the functions.

To solve the question, analyze the given data carefully and write functions accordingly for each plan. Then graph the functions.

Write functions for each plan that give the cost to rent x movies or disks per month for each plan.

For plan A, the cost function is A(x)=2x,

For plan B, the cost function is B(x)=20,

For plan C, cost function is C(x) = 10 + 2.5x

Plot the above functions on the graph.

A(x) = 2x

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 60 Exercise 1 Answer Image 1

C(x) = 10 + 2.5x

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 60 Exercise 1 Answer Image 2

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 60 Exercise 1 Answer Image 3

 

Thus, cost functions for

plan A is A(x) = 2x

plan B is B(x) = 20

plan C is C(x) = 10 + 2.5x,

which were solved by forming cost functions for the cost of x items.

The graph of the functions is

Envision Algebra 1 Assessment Chapter 4 Practice Test C Page 60 Exercise 1 Answer Image 3

In the question, it is given that a person named Kayden is considering three different ways to rent movies and video games.

Rent movies and game disks for $2 each.

Unlimited games and movies to any device for $20 per month.

Rent movies and game disks for a $10 monthly fee plus $2.5 per disk.

Also from the result of the previous part, the costs functions for

plan A is A(x) = 2x

plan B is B(x) = 20

plan C is C(x) = 10 + 2.5x

It is required to tell under what conditions Plan B is better than plan A if only cost is considered.

To solve the question, analyze the given data carefully. Now plan B is better than A if the cost of B is less than A. Solve the inequality and get the conditions using suitable mathematical operations.

Analyze the given data carefully.

For plan A, the cost function is A(x) = 2x,

For plan B, the cost function is B(x) = 20,

Now, the Cost of plan B is less than plan A when 20 < 2x

Divide by 2 on both sides.

\(\frac{20}{2}<\frac{2 x}{2}\)

x > 0

Now, if the number of disks is more than 10, plan A will be more costly than plan B where cost is constant.

Thus, when the number of disks is more than 10, plan B is better than plan A.

Thus, when the number of disks is more than 10, plan B is better than plan A which was solved by analyzing data and using appropriate mathematical operations.

In the question, it is given that a person named Kayden is considering three different ways to rent movies and video games.

Rent movies and game disks for $2 each.

Unlimited games and movies to any device for $20 per month.

Rent movies and game disks for $10 monthly fee plus $2.5 per disk.

Also from the result of the previous part, the cost functions for

plan A is A(x) = 2x

plan B is B(x) = 20

plan C is C(x) = 10 + 2.5x

It is required to tell under what conditions plan C is better than plan B, if only cost is considered.

To solve the question, analyze the given data carefully. Now plan C is better than B if the cost of C is less than B. Solve the inequality and get the conditions using suitable mathematical operations.

Analyze the given data carefully.

For plan B, the cost function is B(x) = 20,

For plan C, cost function is C(x) = 10 + 2.5x

The cost of plan C is less than plan B if 10 + 2.5x < 20

B(x)=20
c(x)=10+2.5x
10+2.5x<20
subtract 10 on both sides

10-10+2.5x<20-10
2.5x<10

Divide both sides by 2.5

\(\begin{gathered}
\frac{2.5 x}{2.5}<\frac{10}{2.5} \\
x<4
\end{gathered}\)

Now, if the number of disks is less than 4, plan C is less costly than plan B.

Thus, when the number of disks is less than 4, plan C is better than plan C which was solved by analysing data and using appropriate mathematical operations.

In the question, it is given that a person named Kayden is considering three different ways to rent movies and video games.

Rent movies and game disks for $2 each.

Unlimited games and movies to any device for $20 per month.

Rent movies and game disks for $10 monthly fee plus $2.5 per disk.

Also from result of previous part, costs functions for

plan A is A(x) = 2x

plan B is B(x) = 20

plan C is C(x) = 10 + 2.5x

It is required to show that Plan A is always more cost effective than Plan C.

To solve the question, analyse given data carefully. Compare costs of two plans by subtracting cost of plan A from plan C. If result is positive that means cost of plan A is always less than C.

Compare costs of plan C and plan A.

For plan A, cost function is A(x) = 2x,

For plan C, cost function is C(x) = 10 + 2.5x

Find their differences.

​C(x) − A(x) = 10 + 2.5x − 2x

= 10 + 0.5x

Now, since x is the number of disks, thus it cannot be negative. This means the difference is always positive.

Solve further for conclusion.

​C(x) − A(x) > 0

C(x) > A(x)

Thus, the cost of plan A is always less than plan C.

Therefore, it is shown that plan A is always cost-effective than plan C which was solved by comparing two costs.

Proof: A(x)=2x
C(x)=10+2.5x
C(x)-A(x)=10+2.5x-2x
=10+0.5x
C(x)-A(x)>0
C(x)>A(x)

Hence showed

In the question, it is given that a person named Kayden is considering three different ways to rent movies and video games.

Rent movies and game disks for $2 each.

Unlimited games and movies to any device for $20 per month.

Rent movies and game disks for $10 monthly fee plus $2.5 per disk.

Also from result of part(a), costs functions for

plan A is A(x) = 2x

plan B is B(x) = 20

plan C is C(x) = 10 + 2.5x

From part (b), plan B is more cost-effective than plan A when the number of disks is more than 10.

As a result of part (d), plan A is always more cost-effective than plan C.

It is required to recommend a plan to Kayden and justify the answer.

To solve the question, analyze the given data and results of previous steps carefully.

Since plan A is always more cost-effective than plan C. Plan B is more cost-effective than plan A when the number of disks is more than 10. It is more likely that he needs less than 10 disks. So, plan A is recommended.

Therefore, plan A) Rent movies and game disks for $2 each, is recommended as it is more likely that Kayden needs less than 10 disks per month.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3

Envision Algebra 1 Assessment Readiness Workbook Chapter 3

Page 42 Exercise 2 Answer

In the question, the given expression is \(\frac{(3)^4(5)^{-3}}{(3)^{-7}(5)^5}\).

It is required to rewrite this expression using positive exponents.

To write the expression using positive exponents, use the definition of negative exponents to convert the numbers with negative powers into positive powers.

Finally, use the concept of multiplication of two same numbers with different exponents.

Rewrite using positive exponents.

Use the definition of negative exponents \(\frac{1}{(3)^{-7}}\) can be rewritten as (3)7 and (5)-3 can be rewritten as \(\frac{1}{(5)^3}\)

Back- substitute these values in the original expression

⇒ \(\frac{(3)^4(5)^{-3}}{(3)^{-7}(5)^5}=\frac{(3)^4(3)^7}{(5)^3(5)^5}\)

Back -substitute these values in the original expression.

⇒ \(\frac{(3)^4(5)^{-3}}{(3)^{-7}(5)^5}=\frac{(3)^4(3)^7}{(5)^3(5)^5}\)

Use the property \(a^m \times a^n=a^{m+n}\)

Add the exponents

⇒ \(\frac{(3)^4(5)^{-3}}{(3)^{-7}(5)^5}=\frac{(3)^{4+7}}{(5)^{3+5}}\)

⇒ \(\frac{(3)^4(5)^{-3}}{(3)^{-7}(5)^5}=\frac{(3)^{11}}{(5)^8}\)

Therefore, the correct option is option \(\text { (A) } \frac{(3)^{11}}{(5)^8}\)

Therefore, the correct option is Option(A) \(\frac{(3)^11}{(5)^8}\).

Option (A) \(\frac{(3)^11}{(5)^8}\) is the positive exponent form of the given expression.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3

Envision Algebra 1 Chapter 3 Answer Key

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 42 Exercise 3 Answer

In the question, the given equation is ∣3x−3∣ + 3 = 9.

It is required to find the solution to this equation.

To find the solution, isolate the absolute value number on one side of the equation and the rest on the other side. Finally, use the relation ∣a∣ = ±a and solve for x.

Apply the definition of absolute value.

Given equation |3x-3|+3=9

Subtract 3 from both sides

|3x-3|+3-3=9-3
|3x-3|=6

Use the definition of absolute value sign

3x-3=±6

Use   the definition of absolute value sign

3x-3=±6

Find the solution 3x-3=6
add 3 on both sides 3x-3+3=+6+3
3x=9
Divide 3 on both sides
⇒ \(\frac{3 x}{3}=\frac{9}{3}\)

x=3

Find the solution 3x-3=-6

Add 3 on both sides

3x-3+3=-6+3

3x=-3

Divide 3 on both sides

⇒ \(\frac{3 x}{3}=\frac{-3}{3}\)

x=-1

Option (B) -1 and Option (C) 3 are the solutions to the given absolute equation.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 42 Exercise 4 Answer

In the question, it is given that a line is perpendicular to a line that passes through (−2,−2) and(2,4).

It is required to find the equation of a line that is perpendicular to the given line.

To find the equation of that line, first, find the slope of the line passing through the given points. Next, use the relation to find the slope of the perpendicular line. Finally, compare the slope of the perpendicular lines from the set of options given.

Find the slope of the line.

The line passes through (-2,-2) and (2,4) using the formula.

⇒ \(m_1=\left(\frac{4-(-2)}{2-(-2)}\right)\)

Simply the signs

⇒ \(m_1=\left(\frac{4+2}{2+2}\right)\)

Add the terms

⇒ \(m_1=\left(\frac{6}{4}\right)\)

Divide the numerator and denominator by 2

⇒ \(\begin{aligned}
& m_1=\frac{\frac{6}{2}}{\frac{4}{2}} \\
& m_1=\frac{3}{2}
\end{aligned}\)

Find the slope of the perpendicular line using the relation of the slope of two perpendicular lines.

m1xm2=-1

Substitute \(\frac{3}{2}\)  for m1

⇒ \(\frac{3}{2} \times m_2=-1\)

multiply by \(\frac{2}{3}\) on both sides

⇒ \(\frac{3}{2} \times \frac{2}{3} \times m_2=-1 \times \frac{2}{3}\)

Simply the equation \(m_2=\frac{-2}{3}\)

Therefore, the slope of the perpendicular line is \(\frac{-2}{3}\).

Write the equation of the perpendicular line.

Use the general equation of a line.

Slope is \(\frac{-2}{3}\).

Substitute \(\frac{-2}{3}\) form in the general equation of a line.

y = \(\frac{-2}{3}\)x + can

Compare with the given options.

Therefore, option (A) y = \(\frac{-2}{3}\)x – 1 is the equation that represents a line perpendicular to the given line.

Option (A) y = \(\frac{-2}{3}\)x – 1 is the equation that represents a line perpendicular to a line that passes through the points (-2, -2) and (2, 4).

Page 42 Exercise 5 Answer

In the question, it is given that the amount Mai saves after working x hours is given by the equation y = 12.5x + 20.

It is required to find the amount that Mai earns per hour.

To find the amount earned per hour, substitute 1 for x in the given equation and solve for y.

Calculate the amount earned per hour.

Given y=12.5x+20​

mai shares ⇒ y=12.5x+20 after working x hours

Substitute 1 for x.

y=12.5(1)+20

Multiply and the terms

y=12.5+20

y=32.5

Therefore, Mai earns 32.5 per hour.

The amount earned by Mai per hour is 32.5.

Envision Algebra 1 Assessment Readiness Workbook Chapter 3 Solutions

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 42 Exercise 6 Answer

In the question, an expression is given as (2x2y2)4.

It is required to simplify this expression.

To simplify the expression, use the power of a product rule. Next, use the power of a power rule and simplify the expression.

Simplify the expression.

Given expression (2x²y²)4

⇒ \(\left(2 x^2 y^2\right)^4=2^4\left(x^2\right)^4\left(y^2\right)^4\)

Use the power of a power rule.

⇒ \(\left(2 x^2 y^2\right)^4=2^4 x^{4\times2} y^{4 \times 2}\)

Multiply the powers

⇒ \(\left(2 x^2 y^2\right)^4=2^4 x^8 y^8\)

Simply the terms

⇒ \(\left(2 x^2 y^2\right)^4=16 x^8 y^8\)

Use the power of a product rule

Compare with the given options.

Therefore, Option (C) 16x8y8 is the simplified form of the given expression.

The simplified form of (2x2y2)4 is given by Option (C) 16x8y8.

Page 43 Exercise 7 Answer

In the question, the given equation of a function is f(x) = ∣3x + 5∣.

It is required to identify the piecewise-defined function that has the same graph as the given function.

The graph of the function

f(x) = |3x + 5| is symmetric about the line x = \(\frac{-5}{3}\).

A piecewise function with the graph same as f(x) = |3x+5| will also change its sign at x = \(\frac{-5}{3}\).

From the given options, the graph of Option (C)

g(x) = \(\left\{\begin{array}{l}
\cdot 3 x+5 ; x \geq \frac{-5}{3} \\
-3 x-5 ; x<\frac{-5}{3}
\end{array}\right.\)

Changes its sign at \(\frac{-5}{3}\).

Therefore,

g(x) = \(\left\{\begin{array}{l}
\cdot 3 x+5 ; x \geq \frac{-5}{3} \\
-3 x-5 ; x<\frac{-5}{3}
\end{array}\right.\) is the correct answer.

The graph of Option (A) g(x) = \(\left\{\begin{array}{l}
3 x+5 ; x \geq 0 \\
-3 x-5 ; x<0
\end{array}\right.\) changes its sign at x = 0 and symmetric about y-axis.

The graph of Option (B) g(x) = \(\left\{\begin{array}{l}
\cdot 3 x+5 ; x \geq \frac{-3}{5} \\
-3 x-5 ; x<\frac{-3}{5}
\end{array}\right.\) changes its sign at x = \(\frac{-3}{5}\).

The graph of Option (D) g(x) = \(\left\{\begin{array}{l}
3 x+5 ; x \geq 0 \\
-3 x-5 ; x<0
\end{array}\right.\) changes its sign at x = 0 and symmetric about y-axis.

Option (C) g(x) = \(\left\{\begin{array}{l}
\cdot 3 x+5 ; x \geq \frac{-5}{3} \\
-3 x-5 ; x<\frac{-5}{3}
\end{array}\right.\) is the piecewise-defined function that has the same graph as the function f(x) = |3x+5|.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 43 Exercise 9 Answer

In the question, it is given that a student scores 87, 78,94, and 84.

It is required to find the score the student must have on the fifth test to get an average score of 87.

To find the fifth score, take the fifth score as some value x. Next, use the formula to find the average of these five scores.

Finally, find the value of x.

Form an equation to calculate the average.

Take the fifth score as x, i.e., P5 = x.

Consider the formula to find the average of n quantities.

There are five terms and the average is 87.

Substitute 87 for Z and 5 for n.

87 = \(\frac{P_1+P_2+P_3+P_4+P_5}{5}\)

Find the fifth score.

The four scores as 87, 78, 94, and 84respectively.

The four scores as 87,78,94 and 84 respectively.

Substitute p1=87, p2=78, p3=94 and P4=84

Therefore, the fifth score is 92.

The student must earn 92 on the fifth test to get an average score of 87.

Page 43 Exercise 11 Answer

A scatter plot is given in the question.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 1

It is required to find the best-fit line for the plot.

Plot the line y = −x + 4 in the given scatter plot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 2

As y = −x + 4 satisfies most points of the plot. So, option (B) is the correct answer.

For option (A),

Plot y = x + 4 in the given scatter plot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 3

As y = x + 4 does not satisfy most points of the plot. So, option (A) is the incorrect answer.

For option (C)

Plot y = x + 3 in the given scatter plot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 4

As y = x + 3 does not satisfy most points of the plot. So, option (C) is the incorrect answer.

For option (D).

Plot y = −x + 3 in the given scatter plot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 5

As y = −x + 3 does not satisfy most points of the plot. So, option (D) is the incorrect answer.

x+4 satisfies most points of the scatter plot. So, option (B) is the correct answer.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 43 Exercise 11 Answer Image 6

Page 43 Exercise 12 Answer

⇒ \(\frac{3}{2}\), -√7, -2, -√9, \(\frac{2}{3}\), 1 are given.

It is required to order them from least to greatest.

To find the correct order of increasing numbers, convert the irrational numbers to decimals by approximation, and compare the numbers.

Estimate the value of −√7

Since, 4 < 7 < 9

So,√4 < √7 < √9

2 < √7 < 3

Since 7 is a bit closer to 9 than 4.

So a good approximation, for −√7 is −2.6

Simplify the values of other numbers

⇒ \(\begin{aligned}
& \frac{3}{2}=1.5 \\
& -\sqrt{9}=-3 \\
& \frac{2}{3}=0.667
\end{aligned}\)

Compare the obtained values to get the results

Since, -3<-2.6<-2<0.667<1<1.5

So, the obtained results are as follows

The required order of the numbers from least to greatest is as follows.

-3 < -√7 < -2 < \(\frac{2}{3}\) < 1 < \(\frac{3}{2}\)

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 44 Exercise 13 Answer

A dot plot is given in the question.

It is required to find the mean and median of the given dot plot to near the hundredth.

To find the required value, find the frequency of given data and then calculate the values of mean and median.

Express the frequency of the given data.

As 3 has 1 dot their frequency is 1.

Similarly, frequency distribution will be as follows.

3,4,4,5,5,6,7.7.7.7.8,8,8,8,8.

Calculate The mean of the given data

⇒ \(=\frac{3+4+4+5+5+6+7+7+7+7+8+8+8+8+8}{15}\)

⇒ \(=\frac{95}{15}\)

⇒ \(=\frac{19}{3}\)

= 6.333

Estimate the median of the given data.

Since 7 lies in the middle of the data 3,4,4,5,5,6,7.7.7.7.8,8,8,8,8.

So, 7 is the median of the given data.

The mean and the median of the given dot plot are 6.333 and 7 respectively.

Page 44 Exercise 14 Answer

In the question, the area of a rectangle is given by the expression 2x2 − 2x − 12, and the width of the rectangle is given by x−3.

It is required to find the length of this rectangle.

To find the length, first, factorize the area expression into a product of simpler expressions. Next, evaluate the terms of the factorized expression other than x−3

Factorize the expression.

Assume A(x) represents the area expression.

Given expression 2x-2x-12

Take 2 commons from each term

A(x)=2(x²-x-6)

Rewrite -x as 2x-3x

A(x) = 2(x(x+2)-3x-6

Take -3 common from the terms -3x and -6

A(x)=2(x(x+2)-3(x+2))

Take(x+2) common.

A(x)=2 (x+2(x-3)

Observe the factorized expression.

Use the formula for the area of a rectangle.

The width of the rectangle is x−3.

All the terms except x−3 represent the length of this rectangle.

Therefore, length is given by 2(x+2).

The length of the rectangle whose area is given by 2x2 − 2x − 12 is 2(x+2).

Page 44 Exercise 15 Answer

In the question, the given inequality is 3x + 2y < 3.

It is required to graph this inequality.

To create a graphical representation of this inequality, use a graphing utility and draw the graph.

Graph the inequality.

Use a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 44 Exercise 15 Answer

The graph of the inequality 3x + 2y < 3 is –

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 44 Exercise 15 Answer

Page 44 Exercise 16 Answer

In the question, the function is given as f(x) = x2 + 4x + 2.

It is required to draw the graph for the given function.

To do so, find the different values and make the table for them. Then draw the graph by plotting the points and making a smooth curve that passes through those points.

Find the values of the function at different points to plot the graph.

Given function f(x)=x²+4x+2

At x=-4 the value of the equation

f(-4)=(-4)²+4(-4)+2
f(-4)=16-16+2
f(-4)=2

The value of the equation is 2

At x=-3 the value of the function

f(-3)=(-3)²+4(-3)+2
f(-3)=9-12+2
f(-3)=-1

The value of the equation is -1

At x=-2 the value of the function

f(-2)=(-2)²+4(-2)+2
f(-2)=4-8+2
f(-2)=-2

The value of the function is -2
At x=-1the value of the function

f(-1)=(-1)²+4(-1)+2
f(-2)=1-4+2
f(-2)=-1

The value of the function is -1
At x=0 the value of the function

f(0)=(0)²+4(0)+2
f(0)=2
The value of the function is 2

The values for the given function are shown in the table.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 44 Exercise 16 Answer Image 1

The graph for the values is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 44 Exercise 16 Answer Image 2

The graph for the given function is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 44 Exercise 16 Answer Image 2

 

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 45 Exercise 17 Answer

In the question, the statement is given as a ball is thrown directly upward from a height of 25 feet with an initial velocity of 96 feet per second.

Also, the height of the h after t second is given as h = −16t2 + 96t + 25.

It is required to find the time taken by the ball to reach the maximum height. Also, it is required to explain it.

Also, it is required to find the maximum height of the ball. And it is needed to explain it.

To do so, find the time that passes from the start of throwing the ball until it comes to maximum height.

Since h describes the ball’s height above the ground, it is the maximum value when instantaneous velocity is equal to zero.

Find the instantaneous velocity by differentiating the equation of height and equate it to zero. Simplify the equation to get time to reach maximum height.

To find the maximum height, substitute the time in h and simplify it.

Find the time to reach maximum height.

Given h=-16t²+96t+25
Differentiate the given equation concerning t.

h'(t)=-16(2)t+96(1)+0

Hence the maximum height is 169 feet.

The ball will reach the maximum height after 3 seconds.

The maximum height is 169 feet.

Page 45 Exercise 18 Answer

In the question, the values for the function is given as

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 45 Exercise 18 Answer Image 1

It is required to make a scatter plot of the data.

And also it is required to find the equation line that best fits.

To do so, use the points to locate them. Then draw the line through the maximum number of points on a scatter plot balancing about an equal number of points above and below the line.

The graph of the given values.

Plot the given points and join them.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 45 Exercise 18 Answer Image 2

The line that fits scatter plots is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 45 Exercise 18 Answer Image 3

From the graph, the equation of the line is y = 2x.

The scattered plot of the given values is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 45 Exercise 18 Answer Image 4

The line that fits scatter plots is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 45 Exercise 18 Answer Image 5

The equation of the line is y = 2x.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 46 Exercise 19 Answer

The length of the hypotenuse of a right triangle is given as 40 cm.

Also, the length of the one leg is given as 24 cm.

It is required to find the length of the other leg.

Let x be the length of the leg to be determined.

Substitute the values 24 for a, 40 for c, and x for b in the formula of Pythagoras.

Substitute a=24,b=xc=40

Hence the length of the other leg of the given right triangle is 32 cm.

Thus, the length of the other leg of the given right triangle is 32 cm in option (c).

The length of the other leg of the given right triangle is 20 cm

Thus, the length of the other leg in option (A) is not the correct answer.

The length of the other leg of the given right triangle is 28 cm

Thus, the length of the other leg in option (B) is not the correct answer.

The length of the other leg of the given right triangle is 36 cm

Thus, the length of the other leg in option (D) is not the correct answer.

The length of the other leg of the given right triangle is 32cm in option (c).

Page 46 Exercise 20 Answer

The quadratic equation is given as 3x2 + 5x − 5 = −1.

It is required to find the solutions of the given quadratic equation to the nearest hundredth.

To do so, substitute the value 3 for a, 5 for b, and, −4 for c.

Given 3x²+5x-5=1
Add-1 to the equation
3x²+5x-5+1=-1+1
simply above equation
3x²+5x-5+1=0
3x²+5x-4=0

Comparing a given quadratic equation

a=3,b=5,c=-4

substitute the values \(\begin{aligned}
& x=\frac{-5 \pm \sqrt{5^2-4(3)(-4)}}{2(3)} \\
& x=\frac{-5 \pm \sqrt{25+48}}{6} \\
& x=\frac{-5 \pm \sqrt{73}}{6}
\end{aligned}\)

The solutions of the given equation

⇒ \(\begin{aligned}
& x=\frac{-5+\sqrt{73}}{6} \\
& x=0.590 \text { and } \\
& x=\frac{-5-\sqrt{73}}{6} \\
& x=-2.257
\end{aligned}\)

The obtained solutions to the nearest hundredth is x = 0.590 and x = −2.26

Hence options (c) and (d) are correct answers.

From the results of the solution options (a),(b),(e) and (f) are not correct answers

The solutions of the given quadratic equation are options (c) and (d).

Page 46 Exercise 21 Answer

The graph is given as y = 2x − 4.

It is required to describe the given graph as a transformation of the graph y = 2x.

To do so, use the transformation rule. To describe the transformation translate the given graph to a specific unit.

Draw the graph y=x

S substitute x=0 in the equation of the function y=0

The function at x=0 is y=0

Substitute x=-1 in the equation of the function y=-1

The value of the function at c =-1 is y=-1

Substitute x=1 in the equation of the function.

y=1

The value of the function at x=1 is y=1.

From the above different values, the graph is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 21 Answer Image 1

Draw the graph y=2x.

substitute x=o in the equation

From the above different values, the graph is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 21 Answer Image 2

The graph y = 2x stretches the function y = x two times.

Draw the graph y=2x.

Substitute x=0 in the equation of the function y=0

From the above different values, the graph is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 21 Answer Image 3

The graph y = 2x − 4 translates vertically the function

y = 2x down 4 units.

Thus, y = 2x − 4 translates vertically the function

y = 2x down 4 units.

The graph y = 2x − 4 translates vertically the function

y = 2x down 4 units.

The correct option is option (A).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 46 Exercise 22 Answer

In the question, the inequality is given as −4x−3 < 13.

It is required to draw a graph of the inequality.

To do so, solve the inequality x. Then using that value plot the graph.

Solve for x.

Given -4x-3<13
Add 3 on both sides
-4x-3+3<13+3
Simplify the above inequality.

-4x<16

Divide 4 on both sides

⇒ \(\begin{aligned}
-\frac{4 x}{4} & <\frac{16}{4} \\
x & <-4
\end{aligned}\)

Therefore, the range is given as x∈(−4,∞)

The graph of the given inequality is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 22 Answer

The graph of the given inequality is shown below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 22 Answer

Page 46 Exercise 23 Answer

The values for the function is given as

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 46 Exercise 23 Answer

It is required to find the equation of the function from the given options.

To do so, substitute the values of x in the given options and check the values of y.

Compare the values obtained with the table given.

Consider the equation y=-x+2

substitute the value x=-2 in equation

y=-(-2)+2
y=4

Substitute the value x=-1 in question

The value y obtained is equal to the value in the table.

Therefore option (d) is the answer.

Consider the equation y = 2x.

Substitute the value x = −2 in the equation.

​y = 2(−2)

y = −4

The value y obtained is not equal to the value in the table.

Therefore option (a) is not the answer.

Consider the equation y = −2x.

Substitute the value x = −2 in the equation.

Substitute the value x = −1 in the equation.

​y = −2(−1)

y = 2

The value y obtained is not equal to the value in the table.

Therefore option (b) is not the answer.

Consider the equation y = x + 2.

Substitute the value x = −2 in the equation.

​y = −2 + 2

y = 0

The value y obtained is not equal to the value in the table.

Option (d) is the answer.

Page 46 Exercise 24 Answer

In the question, the principal amount is given as $325 and the interest 5 is paid quarterly.

It is required to find the amount to be obtained after 5 years.

To do so, use the amount formula.

Substitute the values $325 for P, 5 for r, 5 for t, and 4 for n in the formula and simplify it.

Find the amount.

⇒ \(A=325\left(1+\frac{5}{100 \times 4}\right)^{4(5)}\)

Simply above equation

⇒ \(A=325\left(1+\frac{5}{400}\right)^{20}\)

Take LCM 400 in the equation

⇒ \(\begin{aligned}
& A=325\left(\frac{400+5}{400}\right)^{20} \\
& A=325\left(\frac{405}{400}\right)^{20}
\end{aligned}\)

Simplify the above equation

A=325(1.0125)20
A=325(1.2820)
A=416.662

Hence the amount obtained after 5 years is $416.662

The amount obtained after 5 years is $416.662.

Envision Algebra 1 Student Edition Chapter 3 Practice Problems

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 47 Exercise 25 Answer

The graph is given as y = \(\sqrt[3]{2 x}\)

It is required to describe the given graph as a transformation of the graph y = \(\sqrt[3]{x}\)

To do so, use the transformation rule. To describe the transformation translate the given graph to a specific unit.

Draw the graph y=\(y=\sqrt[3]{x}\)

When x=-1 the value of the function is

The graph for the different values of the function is given below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 25 Answer

Draw the graph \(y=\sqrt[3]{2 x}\)

When x=-1 the value of the function is \(\begin{aligned}
& y=\sqrt[3]{2(-1)} \\
& y=1.2599
\end{aligned}\)

When x=0 the value of the function is

⇒ \(\begin{aligned}
& y=\sqrt[3]{0} \\
& y=0
\end{aligned}\)

When x=1 the value of the function is

⇒ \(y=\sqrt[3]{2}\)

The graph for the different values of the function is given below.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 25 Answer

The graph y = \(\sqrt[3]{2 x}\) translates a horizontal stretch of the function

y = \(\sqrt[3]{x}\) by 2 units.

Thus, the graph of the function y = \(\sqrt[3]{2 x}\) translates a horizontal compress of the function

y = \(\sqrt[3]{x}\) by 2 units.

Therefore option (B) is the correct answer.

The correct option is option (B).

Page 47 Exercise 26 Answer

In the question, the given system of equations is y = 2x − 3 and y = x + 4.

It is required to find the solution to this system of equations.

To find the solution to these equations, use the method of substitution i.e., substitute one variable as a function of another variable to one of the equations. Next, solve for one variable and substitute its value in any one equation to find the value of the other variable.

Find the value of x.

Consider the equation y=2x-3

substitute x+4 for y

x+4=2x-3

Subtract 4 on both sides

x+4-4=2x-3-4
x=2x-7

Multiply -1 on both sides

-(-x)=-(-7)
x=7

Find the value of y
Consider the equation y=2x-3
Substitute 7 for x

y=2(7)-3
y=14-3
y-11

Therefore, the solution for this system of equations is x = 7 and y = 11.

The solution to this system of equations is x = 7 and y = 11.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 47 Exercise 27 Answer

In the question, the transformation of a parent function y = √x is shown below –

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 27 Answer Image 1

It is required to identify which equation represents this transformation.

y = √x − 4 + 5

Draw the graph of the parent function y = √x using a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 27 Answer Image 2

The graph of the parent function starts from the origin.

Transform this as per the given graph.x → x − 4, as the origin shifts 4 along the positive X-axis, and y → y − 5, as the origin shifts 5 along the positive Y-axis.

y= \(y=\sqrt{x-4}+5\)

Substitute x-4 for x and y-5 for y

⇒ \(y-5=\sqrt{x-4}\)

​Add 5 on both sides

y-5+5= \(\sqrt{x-4}+5\)

y= \(\sqrt{x-4}+5\)

Therefore, Option (B) is the correct answer.

y = √x+4 – 5

Draw the graph of y = √x+4 – 5 using a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 27 Answer Image 3

The y = √x+4 − 5 graph does not resemble the transformed graph.

Therefore, Option (A) is not a correct answer.

y = √x-5 + 4

Draw the graph of y = √x-5 + 4 using a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 27 Answer Image 4

The y = √x−5 + 4 graph does not resemble the transformed graph.

Therefore, Option (C) is not a correct answer.

y = √x−5 + 4

Draw the graph of y = √x−5 + 4 using a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 47 Exercise 27 Answer Image 5

The y = √x+5 − 4 graph does not resemble the transformed graph.

Therefore, Option (D) is not a correct answer.

Option (B) y = √x−4 + 5 is an equation for the given transformation of the graph of the parent function y = √x.

Page 47 Exercise 28 Answer

In the question, it is given that a cylinder has a volume of 648πft3 and a height of 18ft.

It is required to find the diameter of this cylinder.

To find the diameter, use the formula for the volume of a cylinder and substitute the values of height and volume.

Find the radius of the base.

Consider the formula for the volume of a cylinder.

Substitute 648π for v and 18 for h

648π= πr² x 18

Divide by 18π on both sides

⇒ \(\begin{aligned}
\frac{648 \pi}{18 \pi} & =\frac{\pi^2 \times 18}{18 \pi} \\
36 & =r^2
\end{aligned}\)

Take square root on both sides

\(\begin{aligned}
& r=\sqrt{36} \\
& r=6 \mathrm{ft}
\end{aligned}\)

Find the diameter of the cylinder
consider the relation d=2r
Substitute 6 for r
d=2(6)

Therefore, the diameter of the given cylinder is 12 feet.

The diameter of a cylinder that has a volume of 648πft3 and a height of 18ft is 12 feet.

Page 48 Exercise 31 Answer

In the question, a set of ages of the members of a sailing club is given as follows –

18,19,24,28,33,37,42, and 46.

It is required to find the change in the mean, median, mode, and range of the ages of the club if a man 42 years old joins the club.

To find the change in the mean, median, mode, and range of the ages of the club, first, find these values for the given dataset.

Next, find these values by adding 42 of them to this dataset. Finally, compare the two values to find the change.

Find the change in the mean.

There are 8 age values at the beginning.

Use the definition of mean.

​Mean1 \(=\frac{18+19+24+28+33+37+42}{8}\)

Add the terms

Mean1= \(\frac{247}{8}\)

Divide the numerator by denominatortor

Mean1 =30.88

Add 42 and recalculate the mean

⇒\(\text { Mean }_2=\frac{18+19+24+28+33+42+46+42}{9}\)

Add the terms

⇒ \(\text { Mean }_2=\frac{289}{9}\)

⇒ \(\text { Mean }_2=\frac{289}{9}\)

Divide the numerator by denominator

Mean2=32.11

The change in  the mean is

Mean2-mean1 = 32.11-30.88

= 1.23

Therefore, in general, the mean of this data set increases by 1.23.

Find the change in the median.

Arrange the terms in increasing order.

18,19,24,28,33,37,42,46

The middle values are 28 and 33.

Use the definition of the median for an even number of terms.

Median1 = \(\frac{28+33}{2}\)

Add the terms.

Median1 = \(\frac{61}{2}\)

Divide the numerator by denominator.

Median1 = 30.5

Add 42 to this set of values and recalculate the median.

Arrange the terms in increasing order.

18,19,24,28,33,37,42,42,46

The middle term is 33.

Use the definition of the median for an odd number of terms.

Median2 = 33

The change in the median is –

​Median2 − Median1 = 33 − 30.5

= 2.5

Therefore, in general, the median of this dataset increases by 2.5.

Find the mode change.

Use the definition of mode.

None of the numbers are repeating in the data set.

No mode for the list at the beginning.

Add 42 to this set of values and recalculate the median.

18,19,24,28,33,37,42,42,46

42 is the only repeating value in this data set.

The mode of the new dataset is 42.

Therefore, in general, the data set changes from no mode value to 42 as the new mode value.

Find the change in range.

The maximum and minimum value in the given data set is 46 and 18 respectively.

Use the definition of the range.

Range = 46 − 18

Subtract the terms.

Range1 = 28

Add 42 to this set of values and recalculate the median.

The maximum and minimum value remains the same.

Therefore, no change in the range of the ages.

The mean of the ages of the members of a sailing club increases by 1.23 when a new member 42 years old join the club.

The median of the ages of the members of a sailing club increases by 2.5 when a new member 42 years old join the club.

The mode of the ages of the members of a sailing club changes from no mode to a value 42 when a new member 42 years old join the club.

The range of the ages of the members of a sailing club remains unchanged when a new member 42 years old join the club..

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 48 Exercise 32 Answer

In the question, an expression is given by 8x4 − 4x3 − 24x2.

It is required to find the factored form of this expression.

To find the factored form, take common variables and numbers in one parenthesis and the rest in another. Finally, form a product of two simplest forms of expressions.

Write the factored form.

Consider the given expression.

Take a common 4x2 from each term of the expression.

Given 8x4-4x3-24x2
Take common 4x2
4x2(2x2-x-6)

Rewrite -x as -4x+3x
4x2(2×2-4x+3x-6)
take common 2x and 3

Take (x-2) Common

4x2 (x-2)(2x+3)

Therefore, the factored form of the given expression is 4x2(x−2)(2x+3).

The factored form of the expression 8x4 − 4x3 − 24x2 by taking common terms outside of the parenthesis and rewriting the quadratic equation as a product of two simple expressions is given by 4x2(x−2)(2x+3).

Page 48 Exercise 33 Answer

In the question, it is given that for an obtuse triangle the measure of the angle A is twice the measure of an angle B and the measure of the angle C is 30° greater than the measure of the angle B.

It is required to find the measure of all three angles.

To find the measure of all three angles, use the concept of the sum of measures of the angles inside a triangle. Next, apply the conditions to make the equation with only one unknown value. Finally, find the one unknown value and solve for the rest of the angles.

Find the measure of the angle B.

Use the concept of the sum of the measures of a triangle.

M∠A+M∠B+M∠C=180°

Apply the given conditions

M∠A=2m∠B and
M∠C=m∠B+30°

Substitute 2m∠B for m∠A and m∠B +30° for m∠c.

2m∠B+m∠B+m∠B+30°=180°

Subtract 30° from both sides
2m∠B+m<B+m<B+30°-30°=180°-30°

2m∠B+m∠B+m∠B=150°

4m∠B=150°

Divide by 4 on both sides

\(\frac{4 m \angle B}{4}=\frac{150^{\circ}}{4}\)

∴ The measure of the angle B is 3.75°

Find the measure of the angle A.

The measure of the angle A is two times the measure of the angle B.

Substitute 37.5∘ for m∠B.

m∠A = 2m∠B
m∠A = 2(37.5)

Multiply the terms.

m∠A = 75°

Therefore, the measure of the angle A is 75°.

Find the measure of the angle C.

The measure of the angle C is 30° greater than the measure of the angle B.

Substitute 37.5° for m∠B.

​m∠C = m∠B + 30°

m∠C = 37.5° + 30°

Add the terms.

m∠C = 67.5°

Therefore, the measure of the angle C is 67.5°.

The measure of the angle is 75°.

The measure of the angle B is 37.5°.

The measure of the angle C is 67.5°.

Page 48 Exercise 34 Answer

In the question, the given function is f(x) = 4x2 − 10x − 6.

It is required to find the zeros of this function.

To find the zeros of the given function, first, equate it zero. Next, find the type of equation and determine the roots.

Find the zeros of the function.

Use the definition of zeros of the function.

Equate the given function to zero.

Given f(x)= 4x²-10x-6

⇒ \(x=\frac{10 \pm \sqrt{(10)^2-4(4)(-6)}}{2(4)}\)

⇒ \(x=\frac{10 \pm \sqrt{(10)^2+4(4)(6)}}{8}\)

⇒ \(x=\frac{10 \pm \sqrt{100+96}}{8}\)

⇒ \(x=\frac{10 \pm \sqrt{196}}{8}\)

⇒ \(x=\frac{10 \pm 14}{8}\)

Therefore, the solutions are

x = \(\frac{10+14}{8}\) and x = \(\frac{10-14}{8}\)

x = \(\frac{24}{8}\) x = \(\frac{-4}{8}\)

x = 3 x = -0.5

Therefore, the zeros of the function are 3 and −0.5.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 3 Page 49 Exercise 35 Answer

In the question, it is given that Dalia has a combination of 24 nickels and quarters for a total of $2.60.

It is required to write a system of equations and determine the number of each type of coin Dalia has.

To write the system of equations, take the number of nickels as x and the number of quarters as $y$ and apply the conditions.

Next, find the number of nickels and quarters by solving this system of equations.

Write the system of equations.

Take the number of nickels as x and the number of quarters as y.

Dalia has a combination of 24 nickels and quarters.

Use this condition and write the first equation.

x + y = 24

1 nickel equals$0.05 and 1 quarter equals $0.25.

Dalia has a total of $2.60.

The sum of x times $0.05 and y times $0.25 will be $2.60.

Use this condition and write the second equation.

0.05x + 0.25y = 2.60

Therefore, the system of equations is x + y = 24 and 0.05x + 0.25y = 2.60.

Find the number of each type of coin.

Use the system of equations from the previous step.

Consider the equation 0.05+0.25y=2.60

Substitute 24-y in x

0.05(24-y)+0.25y=2.60

Simply the parenthesis

1.20-0.05y+0.25y=2.60

Subtract 1.20 from both sides

1.20-0.05y-1.20+0.24y=2.60-1.20-0.05y+0.25y=1.40

The system of equations to determine the type of each coin Dalia has is given by –

x + y = 24 and 0.05x + 0.25y = 2.60

Dalia has 17 nickels and 7 quarters for a total of $2.60.

How To Solve Envision Algebra 1 Chapter 3 Questions

Page 49 Exercise 36 Answer

In the question, the given piecewise-defined function is –

f(x) = \(\left\{\begin{array}{l}
\frac{3}{2} x-4 ; x<0 \\
\frac{1}{3} x-4 ; 0 \leq x<6 \\
2 x-10 ; x>6
\end{array}\right.\)

It is required to graph this function.

To create a graphical representation of this piecewise-defined function, use a graphing utility and draw the graph.

Graph the given function.

Use a graphing calculator.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 49 Exercise 36 Answer

The graph of the given piecewise-defined function is –

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 49 Exercise 36 Answer

In the question, the given piecewise-defined function is –

f(x) = \(\left\{\begin{array}{l}
\frac{3}{2} x-4 ; x<0 \\
\frac{1}{3} x-4 ; 0 \leq x<6 \\
2 x-10 ; x>6
\end{array}\right.\)

It is required to find the domain and range of this function.

To find the domain of this function, observe the graph of the function from the previous part of this problem and write all the possible inputs for x. Next, to find the range of this function, check for all the possible values of f(x).

Find the domain.

Observe the graph from the previous part of this problem.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 49 Exercise 36 Answer

The function is stretched for all the values of x except 6.

Therefore, the domain of the function is (−∞,6)∪(6,∞).

Find the range.

Observe the graph from the previous part of this problem.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 49 Exercise 36 Answer

The function is undefined for the interval (−2,2).

Therefore, the range of the function is (−∞,−2)∪(2,∞).

The domain of the given piecewise-defined function is (−∞,6)∪(6,∞).

The range of the given piecewise-defined function is (−∞,−2)∪(2,∞).

In the question, the given piecewise-defined function is –

f(x) = \(\left\{\begin{array}{l}
\frac{3}{2} x-4 ; x<0 \\
\frac{1}{3} x-4 ; 0 \leq x<6 \\
2 x-10 ; x>6
\end{array}\right.\)

It is required to determine the interval for increasing or decreasing of this function.

To find the interval of increasing or decreasing, check for the slope of the line for each interval.

Check the function for x < 0.

The piecewise-defined function for this interval is f(x) = \(\frac{3}{2}\)x – 4.

Compare with the general equation of a line.

m = \(\frac{3}{2}\).

The slope of the line represented by this function is positive.

Therefore, the function is increasing for the interval (−∞,0).

Check the function for 0 ≤ x < 6.

The piecewise-defined function for this interval is f(x) = \(\frac{1}{3}\)x – 4.

Compare with the general equation of a line.

m = f(x) = \(\frac{1}{3}\).

The slope of the line represented by this function is positive.

Therefore, the function is increasing for the interval [0,6).

Check the function for x > 6.

The piecewise-defined function for this interval is f(x) = 2x − 10.

Compare with the general equation of a line.

m = 2.

The slope of the line represented by this function is positive.

Therefore, the function is increasing for the interval (6,∞).

The given piecewise-defined function is increasing for the interval (−∞,6)∪(6,∞).

Envision Algebra 1 Chapter 3 Step-By-Step Solutions

Page 50 Exercise 1 Answer

In the question, it is given that a high school rectangular parking lot 400ft long and 250ft wide is expanded to double the area by increasing both its length and width by (x)ft.

It is required to draw a diagram that represents the original rectangular parking lot as well as the expanded parking lot with all the labeling.

To draw a rectangular parking lot, first, draw a rectangle that measures 400ft by 250ft. Next, draw a larger parking lot around the original such that the length and the width of the new rectangle are (x)ft more.

Draw the original parking lot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 50 Exercise 1 Answer Image 1

Draw the expanded parking lot.

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 50 Exercise 1 Answer Image 2

The diagram of the original parking lot is –

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 50 Exercise 1 Answer Image 1

The diagram of the expanded parking lot is –

Envision Algebra 1 Assessment Chapter 3 Practice Test B Page 50 Exercise 1 Answer Image 2

In the question, it is given that a high school rectangular parking lot 400ft long and 250ft wide is expanded to double the area by increasing both its length and width by (x)ft.

It is required to form an equation that can be used to find x and solve the equation. It is also required to round the answer to the nearest 10ft.

To form an equation, first, find the area of the original parking lot using the formula for the area of a rectangle. Next, increase the length and the width by x and find the area equation for the expanded parking lot.

Finally, double the area substitute this value in the area equation and find the value of x.

Find the area of the original parking lot.

The length is 400ft and the width is 250ft.

Use the formula for the area of a rectangle.

A=lxb
Substitute l=400
b=250
A=400×250
A=100000ft²

Therefore, the area of the original parking lot is 100000 square feet.

Form the equation for the expanded parking lot.

The length and the width are increased by (x)ft.

The new length of the parking lot is (400+x)ft and the width is (250+x)ft.

Use the formula for the area of a rectangle.

Therefore, the area of the original parking lot is given by A = x2 + 650x + 100000.

Double the original parking lot area.

The area of the original parking lot is 100000ft2.

The area of the expanded parking lot is twice this area.

The area of the expanded parking lot is twice this area

Multiply 100000 by 2

A= 100000×2

A=200000

Substitute A= 20000=x2+650x+100000

Subtract 200000 from both sides

x2+650x+100000-2000000= 100000- 200000

x2+650x+100000=0

Therefore, the equation for the area of the expanded parking lot is given by x2 + 650x − 100000 = 0.

Solve the equation.

Compare the equation with the general form of a quadratic equation.

a=1
b=650
c=-100000

Use the formula for the roots of a quadratic equation

⇒ \(x=\frac{-650 \pm \sqrt{(650)^2-4(-100000)}}{2}\)

Multiple The terms -4 and -100000

⇒ \(x=\frac{-650 \pm \sqrt{(650)^2+400000}}{2}\)

Simply the square root

⇒ \(x=\frac{-650 \pm 906.92}{2}\)

Therefore, the roots of this equation are x = 128.46 and x = −778.46.

Round of the x value.

To increase the size of the lot, the only positive value of x is taken.

Round the value 128.46 to the nearest 10.

Therefore, the value of x is 130ft.

The equation that can be used to find the x value is given by x2 + 650x − 100000 = 0

The value of x after rounding off to the nearest 10 is 130ft.

In the question, it is given that a high school rectangular parking lot 400ft long and 250ft wide is expanded to double the area by increasing both its length and width by (x)ft.

It is required to find the area of the new portion of the expanded parking lot and find the perimeter of the new parking lot.

To find the area of the new portion, first, calculate the area of the expanded parking lot using the value of x from the previous part of the problem. Next, subtract the area of the original parking lot from this new area value. Finally, to find the perimeter of the new parking lot, calculate the new length and width of the rectangle using the formula for the perimeter of a rectangle.

Calculate the area of the new parking lot.

The value of x from the previous part of the problem is 130ft.

The new length and the new width of the rectangle will be 400 + 130 = 530ft and 250 + 130 = 380ft respectively.

Use the formula for the area of a rectangle.

Substitute 530 for l and 380 for b.

A = 530 × 380

Multiply the terms.

A = 201400ft2

Therefore, the area of the expanded parking lot is 201400ft2.

Find the area of the new portion.

The area of the original rectangular parking lot is 100000ft2.

Use the formula for the area of an expanded portion.

Substitute 201400 for Aexp and 100000 for Aorg.

Aportion = 201400 − 100000

Subtract the terms.

Aportion = 101400ft2

Therefore, the area of the new portion is 101400ft2.

Find the perimeter.

The new length and the new width of the rectangle will be 400 + 130 = 530ft and 250 + 130 = 380ft respectively.

Use the formula for the perimeter of a rectangle.

Substitute 530 for l and 380 for b.

P = 2(530+380)

Add the terms inside the parenthesis.

P = 2(910)

Multiply the terms.

P = 1820ft

Therefore, the perimeter is 1820ft.

The area of the new portion of the expanded parking lot is 101400ft2.

The perimeter of the new parking lot is 1820ft.

In the question, it is given that a high school rectangular parking lot 400ft long and 250ft wide is expanded to double the area by increasing both its length and width by (x)ft

It is also given that cost of expanding the lot is $1.50 per square foot of new space and the cost of fencing is $20 per foot.

It is required to find the estimated cost of expanding and fencing in the new lot.

To find the estimated cost of expanding, multiply the cost for per square foot to the area of the new portion of the expanded parking lot. Next, to find the cost of fencing in the new lot, multiply the perimeter of the new lot to the fencing cost for per foot distance.

Calculate the cost of expanding.

The area of the new portion of the expanded lot from the previous part of the problem is 101400ft2.

The cost of expanding the parking lot is $1.50 per square foot of new space.

Multiply 101400 by 1.50.

101400 × 1.50 = 152100

Therefore, the cost of expanding the parking lot $152100.

Calculate the cost of fencing.

The perimeter of the new parking lot from the previous part of the problem is 1820ft.

The cost of fencing is $20 per foot.

Multiply 1820 by 20.

1820 20 = 36400

Therefore, the cost of fencing the new parking lot $36400.

The cost of expanding the parking lot is $152100.

The cost of fencing the new parking lot $36400.

In the question, it is given that a high school rectangular parking lot 400ft long and 250ft wide is expanded to double the area by increasing both its length and width by (x)ft. It is also given that the school has only enough money to pay for half the estimated cost, and thus the parking stickers worth $150 to students and $250to faculties are needed to be sold.

It is required to find the number of student and faculty stickers the school must sell.

To find the number of stickers the school must sell, take the number of student stickers as x and take the number of faculty stickers as y. Next, form an equation with the given condition of the price of stickers and the amount needed for expanding and fencing.

Form an equation.

The total cost of expanding and fencing the lost is 152100 + 36400 = $188500.

Half of the total estimated cost is $94250.

Assume the number of student stickers needed to sell is x and the number of faculty stickers needed to sell is y.

Each student sticker is worth $150 and each faculty sticker is worth $250.

Multiply x by 150 and y by 250.

150x + 250y

Equate 150x + 250y to half of the estimated value.

150x + 250y = 94250

Therefore, the amount raised by selling parking stickers is given by 150x + 250y = 94250.

Find the number of each sticker for one value of x.

The equation to find the number of each sticker is 150x + 250y = 94250.

150(300)+250y=94250

Multiply the terms 150 and 300

45000+250y=94250

Subtract 45000 from both sides

45000-45000+250y=94250-45000

250y=49250

Divide by 250 from both sides

⇒ \(\frac{250 y}{250}=\frac{49250}{250}\)

y=197

Find the number of each sticker for another value of x

The equation to find the number of each sticker is 150x+250y=94250

Take 200 as the value of x.

150(200)+250y =94250
30000+250y=94250

Subtract 30000 from both sides

30000-30000+25y=94250-30000

250y=64250

Divide by 250 from both sides

⇒ \(\frac{250 y}{250}=\frac{64250}{250}\)

y=257

Find the nature of the number of each sticker.

When the value of x increases, the value of y decreases.

When the value of x decreases, the value of y increases.

There are no specific values for the number of student stickers and the number of faculty stickers.

There are multiple possible values for each type of sticker.

There is no specific number of student stickers and faculty stickers that the school must sell to raise half of the estimated cost of the expansion and fencing.

There are multiple possible answers to this problem as the number of student stickers and the number of faculty stickers are relative.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2

Envision Algebra 1 Assessment Readiness Workbook Chapter 2

Page 32 Exercise 1 Answer

It is given that there are five numbers.

a. \(\frac{3}{4}\)

b. √9

c. √12

d. −9

e. √20

It is required to determine all the irrational numbers.

Consider √12. Its value is 2√3 . √3 is an irrational number, hence √12 = 2√3 is an irrational number.

Consider √20. Its value is 2√5 . √5 is an irrational number, hence √20 = 2√5 is an irrational number

Consider \(\frac{3}{4}\). It is a ratio of two integers and hence is a rational number, by definition of rational numbers.

Consider √9. Its value is 3, a rational number.

Consider −9. It is a negative integer and hence a rational number.

Thus option c and option e are irrational numbers.

The correct answer is options c and e, representing irrational numbers √12 and √20.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2

Page 32 Exercise 2 Answer

It is given that the expression is \(\frac{(7)^{-5}(2)^5}{(7)^4(2)^{-2}}\).

a. \(\frac{7^{11}}{2^7}\)

b. \(\frac{7^{9}}{2^3}\)

c. \(\frac{2^{7}}{7^9}\)

d. \(\frac{2^{3}}{7^1}\)

It is required to determine the option that represents the value of the expression using a positive exponent.

Explanation for the correct option:

Option c represents the correct answer.

Consider the given expression

\(\frac{(7)^{-5}(2)^5}{(7)^4(2)^{-2}}\).

The expression can be written in positive exponent using the law of exponent \(.a^{-n}=\frac{1}{a^n}\).

The exponent obtained after applying the law of exponent is \(\frac{(2)^{2}(2)^5}{(7)^4(7)^{5}}\).

Apply the law of exponent ap.aq = ap+q and simplify, \(\frac{(2)^7}{(7)^9}\).

Explanation for other options:

Options a, b, and d do not represent the correct answer because on simplifying, the result obtained is different.

The correct answer is option c, which represents the expression \(\frac{(7)^{-5}(2)^5}{(7)^4(2)^{-2}}\) in terms of positive exponent \(\frac{(2)^7}{(7)^9}\).

Envision Algebra 1 Chapter 2 Answer Key

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 32 Exercise 3 Answer

It is given that the equation is ∣2x−4∣ + 2 = 16. The set of possible solutions is given.

a. −5

b. −4

c. −2

d. 5

e. 9

It is required to determine all possible solutions of the given equation among the given values.

Consider the given equation ∣2x−4∣ + 2 = 16. Subtract both sides by 2 and the simplified equation obtained is ∣2x−4∣ = 14

The solutions of the absolute value are given by the equations 2x − 4 = 14 and 2x − 4 = −14

The value of x is determined from the first equation 2x − 4 = 14, by adding both sides by 4 and then dividing by 2 is 9, given by option e.

The value of x determined from the second equation 2x − 4 = −14, by adding both sides by 4 and then dividing by 2 is −5, given by option a.

Hence the solution of ∣2x−4∣ + 2 = 16 is −5, 9, given by options a and e.

On simplifying the given equation ∣2x−4∣ + 2 = 16 the possible solution set does not match options b, c, d.

The correct answer is option a and e, which represent the solution of the equation ∣2x−4∣ + 2 = 16.

Page 32 Exercise 4 Answer

It is given that a line is perpendicular to another line. The other line passes through the two points (1,0) and (3,−4). Options are given among which one represents the equation of a line perpendicular to the line passing through two points (1,0) and (3,−4).

a. y = 2x + 2

b. y = \(-\frac{1}{2} x+2\)

c. y = 2x + 2

d. y = \(\frac{1}{2} x+2\)

It is required to determine the equation of the line perpendicular to the line passing through two points (1,0) and (3,−4) among the given options.

Option d represents the equation of the line perpendicular to the line passing through two points (1,0) and (3,−4) among the given options. Consider the slope of the line passing through the points (x1,y1) as (1,0) and (x2,y2) as (3,−4). It is given by the formula m = \(\frac{y_2-y_1}{x_2-x_1}\). On substitution the slope is given by \(\frac{-4-0}{3-1}\)

Hence slope of the line through the points (x1, y1) as (1,0) and (x2, y2) as (3, -4) is \(\frac{-4}{2}\) = -2.

The slope of perpendicular lines is negative reciprocals of each other. Hence, the slope of a line perpendicular to the line passing through the points (1,0) and (3,−4) is the negative reciprocal of the slope of −2, that is \(\frac{1}{2}\).

The equation of a line with slope m is y = mx + c with c as the y-intercept. In the given options equation of the line having slope \(\frac{1}{2}\) is y = \(\frac{1}{2} x+2\), that is option d.

The equation formed by a line passing through points (1,0) and (3,−4) does not match options a, b, and c.

The correct answer is option d. y = \(\frac{1}{2} x+2\) is the equation of the line perpendicular to the line passing through two points (1,0) and (3,-4).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 32 Exercise 5 Answer

It is given that the amount of saving after working x hours is given by equation y = 9.5x + 25.

It is required to determine earnings per hour.

Consider the equation for the amount of savings after working x hours, y = 9.5x + 25.

In this equation, the constant 25 represents a fixed amount possibly in the form bonus, some initial payment, or initial saving.

Th.5x represents the amount earned when worked for x hours.

Hence the earning per hour, that is for x as 1, is 9.5 × 1 = 9.5.

The earning per hour is 9.5 as determined from the equation y = 9.5x + 25 which represents the amount of savings after working x hours.

Page 32 Exercise 6 Answer

It is given that the expression is (3×44y2)3.

The four options are given.

a. 3x7y5

b. 3x2y

c. 9x12y6

d. 27x12y6

It is required to determine the option that represents the simplified value of the given expression.

Explanation for the correct option:

The correct answer is represented by option d.

Consider the given expression (3x4y2)3.

The expression can be simplified using the law of exponent (an)m = an.m

The exponent obtained after applying the law of exponent is 27x4.3y2.3 = 27x12y6.

Option d is the option that represents the correct answer.

Explanation for other options:

The options a, b, and c do not represent the correct answer because the simplified equation does not match the result.

The correct answer is option d, which represents the expression (3x4y2)3 in a simplified manner 27x12y6.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 33 Exercise 7 Answer

It is given that the function f(x) = |2x – 3|. The options are given a piecewise function out of which one has the same graph as that of f(x).

a. g(x) = \(\left\{\begin{array}{l}
2 x-3, \text { if } x \geq 0 \\
-2 x+3, \text { if } x<0
\end{array}\right.\)

b. g(x) = \(\left\{\begin{array}{l}
-2 x+3, \text { if } x \geq \frac{2}{3} \\
2 x-3, \text { if } x<\frac{2}{3}
\end{array}\right.\)

c. g(x) = \(\left\{\begin{array}{l}
2 x-3, \text { if } x \geq \frac{3}{2} \\
-2 x+3, \text { if } x<\frac{3}{2}
\end{array}\right.\)

d. g(x) = \(\left\{\begin{array}{l}
2 x-3, \text { if } x \geq 0 \\
2 x+3, \text { if } x<0
\end{array}\right.\)

It is required to determine the piecewise function that has the same graph as f(x).

Explanation for the correct option:

Option c represents a piecewise function that has the same graph as f(x).

Consider f(x) = ∣2x−3∣. The function can be given as two piecewise functions

f1(x) = 2x − 3 and

f2(x) = −(2x−3)

f2(x) = −2x + 3.

The range of the piecewise function can be given by the value of x as determined when y is 0.

From f1(x), 2x − 3 = 0, the value of x is obtained by adding 3 on both sides and dividing both sides by 2, the value of x is \(\frac{3}{2}\).

From f2(x), −2x + 3 = 0, the value of x is obtained by subtracting 3 on both sides and dividing both sides by −2, the value of x is \(\frac{3}{2}\).

The function is 2x−3, with x ≥ \(\frac{3}{2}\).

The function is -2x + 3, with x < \(\frac{3}{2}\).

The function is represented by a piecewise function in option c.

As the functions are the same they have the same graph.

The result obtained when function f(x) = ∣2x−3∣ is simplified does not match options a, b, and d.

The correct option that has the same graph as f(x) = ∣2x−3∣ is given by option c.

Page 33 Exercise 8 Answer

It is given that the function is f(x) = 2x + 5. The given options name different types of equations.

a. exponential

b. linear

c. quadratic

d. none of the above

It is required to determine the equation type of f(x).

The correct answer is option b, linear.

Consider the given equation f(x) = 2x + 5.

The equation is in single independent variable x with power 1.

Hence, the equation f(x) = 2x + 5 represents a linear equation, by definition of linear equation.

An exponential equation is an equation in which the independent variable is an exponent of a constant base.

A quadratic equation is an equation with the highest order of independent variables as 2. In standard form, the quadratic equation is y = ax2+ bx + c.

The correct option is b. The equation f(x) = 2x + 5 is a linear equation.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 33 Exercise 9 Answer

It is given that the scores on the first four tests in one class are 88, 97, 89, 78. The required average score for five tests is 90.

It is required to determine the score to be earned in the fifth test to have an average score of 90.

To determine the score to be earned in the fifth test to have an average score of 90, consider the given scores of four tests and denote the required fifth score as x. Determine the value of x using the formula for average with 5 values,

Average = \(\frac{\text { sum of values }}{\text { number of values }} .\)

Assume the score of the fifth test to be x.

The average of the five test scores is given by the formula Average = \(\frac{\text { sum of values }}{\text { number of values }}.\)

⇒ \(\begin{aligned}
& \text { Average }=\frac{\text { Sum of values }}{\text { number of values }} \\
& 90=\frac{88+97+89+78+x}{5}
\end{aligned}\)

Determine the score of the fifth test x

⇒ \(\begin{aligned}
& 90=\frac{88+97+89+78+x}{5} \\
& 90=\frac{352+x}{5}
\end{aligned}\)

Multiply both sides by 5 and simplify

450=352+x

subtract both sides by 352

450-352=352-352+x

x=98

The score required in the fifth test is 98.

The score to be earned in the fifth test to have an average score of 90 is 98.

Page 33 Exercise 11 Answer

Given: Scatter plot.

It is required to estimate an equation of a line that best fits the given scatter plot.

For this, draw a straight line on the given scatter plot then find the values for m and c from the graph and substitute it in the equation of the straight line.

Draw a straight line that passes through the maximum number of points of a given scatter plot.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 33 Exercise 11 Answer

All the points in the given plot lie between the positive x-axis and the positive y-axis, so the slope of the line is positive. Also, the angle of inclination of the slope with the positive x-axis is less than 90º and the y-intercept is approx 2.

Therefore,

m = 1

c = 2

Substitute 1 for m and 2 for c in the equation y = mx + c.

y = x + 2

Option a represents the line with y-intercept as 3 thus the line moves away from the points.

Option b represents the line with a negative slope so it cannot be the best-fitted line.

Option d represents the line with a negative slope so it cannot be the best-fitted line.

An equation of a line that best fits the given scatter plot is y = x + 2.

Option (C) is correct.

Page 33 Exercise 12 Answer

In the question, the given series is \(\frac{1}{2}\),√7,−√2,−2,\(\frac{3}{4}\),3.

It is required to arrange the given numbers in order from least to greatest.

For this, first, convert the given numbers into integers and then compare them.

Convert numbers other than -2 and 3 into integers

⇒ \(\begin{aligned}
\frac{1}{2} & =0.5 \\
\sqrt{7} & =2.6 \\
-\sqrt{2} & =-1.41 \\
\frac{3}{4} & =0.75
\end{aligned}\)

0.5,2.6-141,0.75

Compare numbers and arrange them in ascending order

-2<-1.41<0.5<0.75<2.6<3
\(-2<-\sqrt{2}<\frac{1}{2}<\frac{3}{4}<\sqrt{7}<3\)

Compare numbers and arrange them in ascending order

The required arrangement of numbers is given as -2 < -√2 < \(\frac{1}{2}\) < \(\frac{3}{4}\) < √7 < 3

Envision Algebra 1 Assessment Readiness Workbook Chapter 2 Solutions

Page 34 Exercise 13 Answer

Given data shown in the dot plot to the nearest hundredth.

It is required to find the mean and median for the given data and tell the reason behind the different values of mean and median.

To find the mean and median it is required to extract data from the given dot plot, then find the mean and the median of the given data using the formula mentioned in the tip section.

Extract and compare information from the given dot plot.

6,6,6,6,7,7,7,7,7,8,8,10,11

Total number of observations is 13.

Substitute required values in the formula,

⇒ \(\begin{aligned}
& \text { Mean }=\frac{\text { Sum of observations }}{\text { total number of observations }} \\
& \text { Mean }=\frac{6+6+6+6+7+7+7+7+7+8+8+10+11}{13}
\end{aligned}\)

Mean =\(\frac{96}{13}\)

Mean =7.38

Here, n is an odd number

Substitute 13 for n in the formula

Median =\(\left(\frac{n+1}{2}\right)^{\text {th }} \text { term }\)

Median = \(\left(\frac{13+1}{2}\right)^{\text {th }} \text { term }\)

Median

Both the values are different because the mean is the average of given numbers and the median is the middle term.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 34 Exercise 14 Answer

Given that the area of the rectangle is 6x2 + 8x + 2 and the length of the rectangle is 3x + 1.

It is required to find the width of the rectangle.

To find the width of the rectangle, substitute the given values in the formula given in the tip section. Factorize the numerator part and then solve it.

Substitute 6x2 + 8x + 2 for A and 3x + 1 for l in formula A = l × b and simplify.

Given 6x²+8x+2 and the length of the rectangle is 3x+1
A=lxb
Substitute A=6x²+8x+2 and l=3x+1
(3x+1)b=6x²+8x+2

⇒ \(b=\frac{6 x^2+8 x+2}{3 x+1}\)

Factorize the numerator of equation \(b=\frac{6 x^2+8 x+2}{3 x+1}\)

⇒ \(b=\frac{2 x(3 x+1)+2(3 x+1)}{3 x+1}\)

​⇒ \(b=\frac{(2 x+2)(3 x+1)}{(3 x+1)}\)

b=2(x+1)

The required value of the width of the rectangle is 2(x+1).

Page 34 Exercise 15 Answer

In the question, the given inequality is 2x + 3y > 12.

It is required to graph the inequality 2x + 3y > 12.

To graph the inequality 2x + 3y > 12, it is required to plot the graph for 2x + 3y = 12 and then compare the points above and below the line

2x + 3y = 12 and shade the region in which the point satisfies the given inequality.

Plot the graph for 2x + 3y = 12.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 15 Answer Image 1

Plot points (1,2) and (3,4) in the graph 2x + 3y = 12 and check the given inequality for both points.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 15 Answer Image 2

For (1,2)

2(1) + 3(2) = 8 < 12

For (3,4)

2(3) + 3(4) = 18 ≥ 12

Shade the region in the graph which satisfies the inequality 2x + 3y > 12.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 15 Answer Image 3

The required graph for inequality 2x + 3y > 12 is given below.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 15 Answer Image 3

Page 34 Exercise 16 Answer

Given function: f(x) = x2 + 2x + 1

It is required to plot the graph for the given function f(x) = x2 + 2x + 1.

For this, it is required to plot a graph for x2 and shift this graph to the left 1 unit to get the required graph for f(x) = x2 + 2x + 1.

Plot the graph for x2.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 16 Answer Image 1

Compare x2 + 2x + 1 with expression (a+b)2 = a2 + 2ab​ + b2.

Thus f(x) = x2 + 2x + 1 can be written as (x+1)2.

f(x) = x2 will shifted to left 1 unit to get f(x) = (x+1)2.

Plot graph for f(x) = x2 + 2x + 1.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 16 Answer Image 2

The required graph for f(x) = (x+1)2 is given below.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 34 Exercise 16 Answer Image 2

Page 35 Exercise 17 Answer

Given that a ball is thrown directly upward from a height of 10ft and an initial velocity of 32ft/s. The given equation h = −16t2 + 32t + 10 gives the height h after t seconds.

It is required to find the maximum height of the ball and the time taken by the ball to reach its maximum height.

For this, use the basic theory of velocity mentioned in the tip section to derive the linear equation and then solve it at the maximum height of the ball to find the time taken by the ball to reach its maximum height. Further, substitute the value of maximum time in the given expression to find the maximum height.

Velocity \(=\frac{d h}{d t}\)

⇒ \(V=\frac{d}{d t}\left(-16 t^2+32 t+10\right)\)

V=-32t+32

At maximum height velocity is zero

0=-32t+32

Solve for t

32t=32

⇒ \(t=\frac{32}{32}\)

t=1 sec

Substitute 1 for t in expression h= -16t²+32t+10

h=-16(1)² +32(1) +10
=-16+32+10
=26ft

The time taken by the ball to reach its maximum height is 1 sec.

The maximum height of the ball is 26ft.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 35 Exercise 18 Answer

Given: A table containing data for 7 points.

It is required to make a scatter plot of the data and estimate an equation of the line of best fit.

To make a scatter plot from the given data, locate the given points in the graph. To estimate the equation of the line, draw a line such that there are equal numbers of points above and below the line at a minimum distance then find x and y intercepts.

Draw a scatter plot.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 35 Exercise 18 Answer Image 1

Draw a line such that there are equal numbers of points above and below the line at minimum distance then find x and y intercepts.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 35 Exercise 18 Answer Image 2

x-intercept is 5.

y-intercept is

Substitute 5 for a and 8 for b in the equation \(\frac{x}{a}+\frac{y}{b}=1\).

⇒ \(\frac{x}{5}+\frac{y}{8}\)=1

An equation of the line of best fit is \(\frac{x}{5}+\frac{y}{8}\)=1.

Page 36 Exercise 20 Answer

Given equation: 2x2 + 7x − 2 = 3

It is required to find the solutions of the equation to the nearest hundredth.

To find the solutions of the equation to the nearest hundredth, use the quadratic formula mentioned in the tip section as the given equation is a quadratic equation.

Explanation for the correct option:

Write the given formula 2x2 + 7x − 2 = 3 in the general form

Given equation: 2x²+7x-2=3
ax²+bx+c=0
2x²+7x-2=3
2x²+7x-2-3=0
2x²+7x-5=0

Substitute a=2, b=7, c=-5 in formula

Option a and e are the correct answers.

Explanation for other options:

Option b, c, d, and f do not match the values of x obtained while solving the given quadratic expression.

The solutions of the equation to the nearest hundredth are −4⋅11 and 0⋅61.

Options (A) and (E) are correct.

Page 36 Exercise 22 Answer

It is given that the inequality −2x + 2 ≤ 8.

It is required to graph the solution of the given inequality.

To graph the solution of the given inequality, find the solution of the inequality using rules for solving inequality, subtract 2 on both sides of the inequality divide by −2 reverse the sign of inequality, and graph the resulting solution.

Consider the given inequality

-2x+2≤8

Subtract 2 on both sides of the inequality

-2x+2-2≤8-2
-2x≤6

Divide by -2 on both sides and reverse the sign

⇒ \(\frac{-2 x}{-2} \geq \frac{6}{-2}\)

x≥-3

So, the graph of the solution will be,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 36 Exercise 22 Answer

Hence, the graph of the solution will be,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 36 Exercise 22 Answer

Page 36 Exercise 23 Answer

It is given that the table of points,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 36 Exercise 23 Answer Image 1

It is required to find which function the table represents.

To plot the points on a graph find the slope of the best-fit line and use the slope-intercept form to find the equation of the line.

Explanation for the correct option:

Plot the points on the graph,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 36 Exercise 23 Answer Image 2

The slope of the best-fit line will be,

The slope of the best-fit line will be,

So the equation of the line in slope-intercept form will be

y=mx+c
y=x+c

Consider the point (o,3) and substitute in the slope-intercept form,

3=0+c
c=3

So, the equation will be,

y=x+3

Explanation for other options:

The equation of the line plotted using the given data does not match the equations of options a, b, and d.

Hence, the correct option will be (c).

Page 36 Exercise 24 Answer

It is given that a $220 deposit in the account that pays 3% interest compounded quarterly.

It is required to find the amount in the account after 5 years.

To find the amount in the account after 5 years, substitute 220 for P, 0.03 for r, 4 for n, and 5 for t in the compound interest formula.

Substitute 220 for P, 0.03 for r, 4 for n, and 5 for t in the formula

\(A=P\left(1+\frac{r}{n}\right)^{n t}\)

Substitute p=220,r=0.03, n=4,t=5

⇒ \(\begin{aligned}
& A=220\left(1+\frac{0.03}{4}\right)^{4 \times 5} \\
& A=220\left(\frac{4.03}{4}\right)^{20} \\
& A=220(1.161) \\
& A=255.42
\end{aligned}\)

Hence, the amount in the account after 5 years will be $255.42.

Page 37 Exercise 25 Answer

It is given that the parent function y = \(\sqrt[3]{x}\).

It is required to describe the graph of y = \(4 \sqrt[3]{x}\) as a transformation of the parent function.

To describe the graph of y = \(4 \sqrt[3]{x}\) as a transformation of the parent function, plot both the curves and observe the transformation.

The graph of the parent function and given function will be,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 25 Answer

It can be observed from the graph that the parent function is stretched vertically by a factor of 4 units.

Option a indicates that the function is translated 4 units up but the graph is not shifted thus this option is not applicable.

Option b indicates that the function is compressed vertically by a factor of 4 but from the graph it is seen that there is no vertical compression thus this option is not applied.

Option d indicates the reflection along the line y = 4 but this is also not correct as the graph of the function is not a straight line.

Hence, the correct option is (c), vertical stretch by a factor of 4.

Envision Algebra 1 Student Edition Chapter 2 Practice Problems

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 37 Exercise 26 Answer

It is given that linear equations are

y = 4x + 1 and

y = 3x + 5.

It is required to find the solution of the system of linear equations.

To find the solution of a system of linear equations equate the right side of both the equations as the left sides of the equations are equal. Then solve the equation for x. Substitute the value of x obtained in one of the given equations and find the value of y.

The left-hand side of both the linear equations is equal thus equation is the right-hand side of the equations.

Given Linear Equations are

y=4x+1 and
y=3x+5
4x+1=3x+5

Subtract both sides of the equation by 3x

4x+1-3x=3x+5-3x
x+1=5

Substitute 4 for x in equation y=4x+1

Thus the solution of a system of linear equations is (4,17).

Page 37 Exercise 27 Answer

It is given that the parent function is y = √x.

It is given that the graph of a transformed function is,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 27 Answer Image 1

It is required to find the equation of transformation of the graph shown in the figure of the parent function y=√x.

To find the equation of transformation of the given graph of the parent function y = √x use the transformation rule :

f(x) + b shifts the function b units upward.

f(x) − b shifts the function b units downward.

f(x+b) shifts the function b units to the left.

f(x−b) shifts the function b units to the right.

Plot the graph of the parent function along with the given graph.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 27 Answer Image 2

From the graph it is seen that graph of the function is shifted 2 units to the left and 3 units upward.

Thus from transition rule f(x+2)+3.

The equation for the transformation of the graph is given by \(\sqrt{x+2}+3\).

The equation for the transformation of the graph is given by \(\sqrt{x+2}+3\).

The correct answer is option (D).

Page 37 Exercise 28 Answer

It is given that the volume of a cylinder is 320π ft3 and the height is 20 ft.

It is required to find the diameter of the base of the cylinder.

To find the diameter of the base of the cylinder substitute the given values of height and volume in formula V = πr2h and solve r where r is the radius of the base. The diameter is twice the radius thus multiply the value of r by 2 to find the diameter.

Given that the volume of a cylinder is 320πft³​ and height is 20ft

v=πr²h

Substitute 320π =v and h = 20

320π =πr²(20)

Divide both sides of the equation

320π = πr²(20) by 20π

⇒ \(\frac{320 \pi}{20 \pi}=\frac{\pi r^2(20)}{20 \pi}\)

16=r²

Rearrange the equation

r²=16

Square root on both sides

⇒ \(\begin{gathered}
\sqrt{r^2}=\sqrt{16} \\
r=\sqrt{4^2}
\end{gathered}\)

Since r is the length of the radius thus it cannot be negative.

Multiply radius r by 2 to find diameter.

D = 2(4)

= 8

The diameter of the cylinder is 8 ft.

Page 37 Exercise 29 Answer

It is given that table shows the preference of students in grades 9th and 10th to listen to music while studying or study in quiet.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 29 Answer Image 1

It is required to complete the table.

It is required to find the percentage of students in grade 10th who prefer listening to music while studying.

To complete the table given simple addition is required. To find the percentage of students in grade 10th who prefer listening to music while studying divide the number of students who prefer listening to music while studying by the total number of students in grade 10 and multiply by 100.

To complete the table perform addition. Add both rows and columns and write the total result in the last row and last column.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 29 Answer Image 2

To find the percentage of students in grade 10th who prefer listening to music while studying divide the number of students that prefer listening to music while studying by the total number of students in grade 10 and multiply by 100.

9 students in 10th grade prefer to listen to music while studying out of 40 students in 10th grade.

Thus percentage can be obtained as,

= \(\frac{9}{40} \times 100\)

= 22.5%

The complete table is,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 37 Exercise 29 Answer Image 2

Percent of 10th graders who listen to music while studying is 22.5%.

Page 37 Exercise 30 Answer

It is given that points and slopes are as follows:

(3,18) and (−2,8) \(\frac{1}{2}\)

(2, -5) and (-4, 7)  2

(12, -3) and (-12, 9) \(\frac{-1}{2}\)

(4,-2) and (12, 6)  -2

It is required to match each pair of points to the slope of the line that passes through these points.

To match each pair of points to the slope of the line that passes through these points use formula m = \(\frac{y_2-y_1}{x_2-x_1}\) to find the slope of each line that passes through a given pair of lines and then match them with the correct slope.

Substitute 8 for y2, 18 for y1 -2 for x2 and 3 for x

⇒ \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m=\frac{8-18}{-2-3} \\
& m=\frac{-10}{-5} \\
& m=2
\end{aligned}\)

The slope of the line that passes through a pair of points (3,18) and (2,8) is 2.

Substitute 7 for y2,-5 for y1-4 for x2 and 2 for x,

⇒ \(\begin{aligned}
& m=\frac{y_2-y_1}{x_2-x_1} \\
& m=\frac{7+5}{-4-2} \\
& m=\frac{12}{-6} \\
& m=-2
\end{aligned}\)

The slope of the line that passes through a pair of points (2,-5) and (-4,7) is -2.

Substitute 6 for y2 x-2 for y1 12 for x2,-4 in formula \(m=\frac{y_2-y_1}{x_2-x_1}\)

⇒ \(\begin{aligned}
m=\frac{y_2-y_1}{x_2-x_1} \Rightarrow m & =\frac{6+2}{12+4} \\
m & =\frac{8}{16} \\
m & =\frac{1}{2}
\end{aligned}\)

Substitute 9 for y2-3 for y1 -12 for x

⇒ \(\begin{aligned}
m=\frac{y_2-y_1}{x_2-x_1} \Rightarrow m & =\frac{9+3}{-12-12} \\
m & =\frac{12}{-24} \\
m & =\frac{-1}{2}
\end{aligned}\)

The slope of the line that passes through a pair of points (12,-3)(-12,9) is \(\frac{-1}{2}\)

The slope of the line that passes through

The slope of the line that passes through a pair of points (3,18) and (−2,8) is 2.

The slope of the line that passes through a pair of points (2,−5) and (−4,7) is −2.

The slope of the line that passes through a pair of points (12,−3) and (−12,9) is \(\frac{1}{-2}\).

The slope of the line that passes through the pair of points (-4, -2) and (12, 6) is \(\frac{1}{2}\).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 38 Exercise 31 Answer

It is given that the age of members of the chess club is 13,15,22,23,35,38,49,56. A new member whose age is 48 years joins the club.

It is required to describe how the mean, median, mode, and range of the ages of club members are affected after joining of new member.

To describe how the mean, median, mode, and range of the ages of club members are affected after joining of new member first find the mean, median, mode, and range of existing club members then find the mean, median, mode, and range of existing members along with new member and see the changes in the values.

Data for old club members: 13,15,22,23,35,38,49,56

The mean is the sum of all the ages divided by no. of members.

Mean ,= \(\frac{13+15+22+23+35+38+49}{8}\)

Mean = \(\frac{251}{8}\)

Mean = 31.375

Since total number of outcomes are even thus median is given by formula \(\frac{\left(\frac{n}{2}\right)^{t h} \text { term }+\left(\frac{n}{2}+1\right)^{t h} \text { term }}{2}\) where n the total number of outcomes.

Median = \(\frac{\left(\frac{8}{2}\right)^{\text {th }} \operatorname{term}+\left(\frac{8}{2}+1\right)^{\text {th }} \text { term }}{2}\)

Median= \(\frac{4^{\text {th }} \text { term }+5^{\text {th }} \text { term }}{2}\)

From the data, it is seen that the fourth term is 23 and the fifth term is 35 thus simplifying to obtain the median.

median = \(\frac{23+35}{2}\)

= \(\frac{58}{2}\)

= 29

Mode for the given data does not exist because each value occurs only once in the data.

The range of the data is the highest value subtracted by the lowest value of the data set.

From the data, it is seen that the highest value is 56 and the lowest value is 13 thus,

Range = 56 − 13

Data for old club members along with new joining: 13,15,22,23,35,38,48,49,56

The mean is the sum of all the ages divided by no. of members.

mean = \(\frac{13+15+22+23+35+38+48+49+56}{9}\)

= \(\frac{251}{9}\)

= 33.23

since the total number of outcomes is odd the median is given by the formula \(\left(\frac{n+1}{2}\right)^{t h} \text { term }\) where n is the total number of outcomes.

median = \(\left(\frac{9+1}{2}\right)^{t h} \text { term }\)

= 5th term

From the data it is seen that the fifth term is 35 thus median will be 35.

Mode for the given data does not exist because each value occurs only once in the data.

The range of the data is the highest value subtracted by the lowest value of the data set.

From the data, it is seen that the highest value is 56 and the lowest value is 13 thus,

Range = 56 − 13

From the above data, it is concluded that the mean and median are increased after joining of new member whereas the mode and range of the data remain the same.

It is concluded that the mean and median are increased after joining of new member whereas the mode and range of the data remain the same.

Page 38 Exercise 33 Answer

It is given that the measure of angle A is three times the measure of angle B and the measure of angle C is one-half times the measure of angle B.

It is required to find a measure of all the three angles.

To find a measure of all three angles use the triangle sum property. The sum of all three angles of an obtuse triangle is 180°.

Let the measure of angle B be x.

Use the triangle sum theorem to form an equation as the measure of angle A is three times the measure of angle B and the measure of angle C is one-half times the measure of angle B.

Solve the expression

⇒ \(x^0+3 x^{\circ}+\frac{1}{2} x^0=180^{\circ}\)

Mathematical operations and find the value of x

⇒ \(\begin{aligned}
& x^{\circ}+3 x^{\circ}+\frac{1}{2} x^{\circ}=180^{\circ} \\
& \frac{9}{2} x^{\circ}=180^{\circ}
\end{aligned}\)

Multiply both sides of the expression

⇒ \(\begin{aligned}
& \frac{9}{2} x^{\circ}=180^{\circ} \text { by } \frac{2}{9} \\
& \frac{9}{2} x^{\circ} \times \frac{2}{9}=180^{\circ} \times \frac{2}{9}
\end{aligned}\)

x°=40°

Thus the measure of angle B is 40°.

To find a measure of angle A multiply the measure of B by 3 and simplify.

= 3(40°)

= 120°

​To find a measure of angle C divide the measure of B by 2 and simplify.

= \(\frac{40^{\circ}}{2}\)

= 20°

The measure of angle A is 120°, the measure of angle B is 40°, and the measure of angle C is 20°.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 38 Exercise 34 Answer

It is given that the function is f(x) = 2x2 − 6x + 4.

It is required to find zeros of the given function.

To find zeros of the function equate the function to 0. Then split the middle term into two terms such that the sum of those two terms is equal to the middle term and the product of those two terms is equal to the first and last term.

Equate the function f(x) = 2x2 − 6x + 4 to 0.

Given f(x)= 2x²-6x+4=0

2x²-2x-4x+4=0
2x(x-1)-4(x-1)=0
(2x-4)(x-1)=0

Either (2x-4) is equal to o or (x-1) is equal to 0

Let 2x-4=0

Add 4 on both sides

2x-4+4=0+4
2x=4

Divide both sides into 2

⇒ \(\begin{aligned}
\frac{2 x}{2} & =\frac{4}{2} \\
x & =2
\end{aligned}\)

Let x=1=0

Add 1 on both sides
x-1+1=0+1
x=1

Zeros of the function f(x) = 2x2 − 6x + 4 are 2 and 1.

Page 39 Exercise 35 Answer

It is given that total number of coins is 30 and the total cost is $2.10.

It is required to write a system of equations and find the number of each type of coin.

To write a system of equations use the given condition and fact that a nickel worth $0.05 and a dime worth $0.10. Then solve the two equations to find the number of each type of coin.

Let x be the number of nickel coins and y be the number of dime coins.

By the given condition that the total number of coins is 30 the equation formed is x + y = 30.

Then equation is formed by another condition total cost is $2.10 and the fact that a nickel is worth $0.05 and a dime worth $0.10 is 0.05x + 0.10y = 2.10.

Given 0.05+0.10y=2.10-1
rearrange equation x+y=30
y=30-x

Substitute y =30-x in the equation

The system of equations used to determine the number of each type of coin are:

x + y = 30 and

0.05x + 0.10y = 2.10

There are 18 nickel coins and 12 coins of dime.

Envision Algebra 1 Chapter 2 Step-By-Step Solutions

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 2 Page 39 Exercise 36 Answer

It is given that the piecewise function is:

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\)

It is required to plot the given piecewise function.

Graph the piece-wise function

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\) where

the x-axis represents different values of x and the y-axis represents the values of the function obtained while substituting the values of x.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 39 Exercise 36 Answer Image 1

The graph of the piece wise function f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\) is as follows,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 39 Exercise 36 Answer Image 1

It is given that the piecewise function is

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\)

It is required to find the domain and range of the function.

To find the domain and range of the function use the definition of range and domain.

The first condition of function shows that it is given that function exists at every point x < −2.

The second condition shows that the function exists at point −2 also.

The third condition shows that the function exists at every point where x>4.

This domain can be written as (−∞,4)∪(4,∞).

The range is a set of all output values. Thus range of functions is the entire real line.

Domain of the function

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\) is

(−∞,4)∪(4,∞) and range is (−∞, ∞).

It is given that the piecewise function is

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\).

It is required to find the interval on which the function is increasing or decreasing.

To find the interval on which function is increasing or decreasing see the graph of the function and use the definition of increasing and decreasing functions to find the interval.

The graph of the function

f(x) = \(\left\{\begin{array}{l}
2 x-2 \text { if } x<-2 \\
1.5 x \text { if }-2 \leq x<4 \\
2 \text { if } x>4
\end{array}\right.\) is,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 39 Exercise 36 Answer Image 2

It is concluded from the graph that the value of the function decreases as the value of x decreases over the interval (−∞,−2).

It is concluded from the graph that the value of the function is increasing as the value of x increases over the interval [−2,4).

It is concluded that over the interval (4,∞) the value of the function remains constant as the value of x increases over the interval (-2, 4).

It is concluded that over the interval (4, ∞) the value of the function remains constant as the value of x increases.

It is concluded from the graph that the value of the function is decreasing as the value of x decreases over the interval (−∞,−2).

It is concluded from the graph that the value of the function is increasing as the value of x increases over the interval [−2,4).

It is concluded that over the interval (4,∞) the value of the function remains constant as the value of x increases over the interval (-2, 4).

It is concluded that over the interval (4, ∞) the value of the function remains constant as the value of x increases.

How To Solve Envision Algebra 1 Chapter 2 Questions

Page 40 Exercise 1 Answer

It is given that the initial height of a baseball is 5ft, and the initial speed is 60ft/s.

It is required to find an equation that represents the ball’s path and graph the equation.

To find an equation that represents the balls path and graph the equation substitute the given values in the formula y = \(-\frac{g}{V^2} x^2+x+y_0\) and simplify to form an equation in x and y. Then use the graphing tool to plot the graph of the function.

Substitute 5 for y0, 60 for V and 32 for g in formula y = \(-\frac{g}{V^2} x^2+x+y_0\) and simplify.

y = \(-\frac{32}{(60)^2} x^2+x+5\)

y = -0.0089x2 + x + 5

plot the graph of a function

y = -0.0089x2 + x + 5 where the x-axis represents the horizontal distance traveled by the baseball and the y-axis represents the height of the baseball.

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 40 Exercise 1 Answer Image 1

The graph of the function along with Felipe’s position is,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 40 Exercise 1 Answer Image 2

From the graph it is seen that ball passed the Felipe’s position thus the ball did not land in Felipe’s gloves.

The equation that represents the ball path is y = −0.0089x2 + x + 5.

Graph that represents the equation y = −0.0089x2 + x + 5 is as follows,

Envision Algebra 1 Assessment Chapter 2 Practice Test A Page 40 Exercise 1 Answer Image 3

From the graph it is seen that ball passed the Felipe’s position thus the ball did not land in Felipe’s gloves.

It is given that the initial height of a baseball is 5ft, and the initial speed is 60ft/s.

It is required to find an equation that represents the ball’s path and graph the equation.

To find an equation that represents the ball path and graph the equation substitute the given values in the formula y = \(-\frac{g}{V^2} x^2+x+y_0\) to find the value of x. Thus the point obtained is the point that lies on the ball’s path.

⇒ \(y=\frac{-9}{v^2} x^2+x+y_0\)

Substitute y=5ft,y0=5ft,v=60ftls,g=32ftls²

⇒ \(5=-\frac{32}{(60)^2}\left(x^2\right)+90+5\)

⇒ \(\begin{aligned}
& 0=\frac{-32}{3600} \times x^2+x \\
& 0=-\frac{2 x^2}{225}+x
\end{aligned}\)

Rearrange equation \(\frac{2 x^2}{225}-x=0\)

2x²-225x=0

Take x as a common factor

x(2x-225)=0

Let x = 0.

Thus one point that lies on the ball’s path so that Felipe catches the ball is(0,5) but this is not possible as the ball does not travel any horizontal distance.

Let 2x-225=0

Add 225 on both sides

2x-225+225=0+225

Divide 2 on both sides

⇒ \(\frac{2 x}{2}=\frac{225}{2}\)

x=112.5

Thus the point that lies on the ball’s path so that Felipe catches the ball is (112.5,5).

The points that lie on the ball’s path so that Felipe catches the ball is (112.5,5) because it is given that Felipe is 5ft above the ground so the value of y is substituted as 5ft in the formula y = \(-\frac{g}{V^2} x^2+x+y_0\) with given initial conditions.

It is given that the answer to part b is to be used.

It is required to find an equation that describes the initial height and initial speed for which a friend catches the ball.

To find an equation that describes the initial height and initial speed for which a friend catches the ball substitute the point (112.5,5) for (x,y) in

\(y=\frac{-g}{v^2} x^2+x+y_0\)

Substitute (112.5,5) for(x,y)

⇒ \(5=\frac{-9}{v^2}(112.5)^2+(112.5)+y_0\)

⇒ \(\begin{aligned}
& 5-112.5=\frac{-405000}{v^2}+y_0 \\
& -107.5+\frac{405000}{v^2}=y_0
\end{aligned}\)

Thus the equation that describes initial height and initial speed is

⇒ \(y_0=\frac{405000}{V^2}-107.5\).

The equation that describes initial height and initial speed is \(y_0=\frac{405000}{V^2}-107.5\).

There is only one possible equation to describe the initial height and initial speed of the ball.

It is given that a friend catches the ball.

It is required to find the initial height and initial speed for which a friend catches the ball.

To find the initial height and initial speed for which a friend catches the ball, use the hit and trial method and substitute values of V to find values of y0 in the equation

⇒ \(y_0=\frac{405000}{V^2}-107.5\) and check which value is possible.

Let v=90

Substitute 90 for v in the equation

⇒ \(\begin{aligned}
y_0 & =\frac{405000}{v^2}-107.5 \\
y_0 & =\frac{405000}{(90)^2}-107.5 \\
& =50-107.5 \\
& =-57.5
\end{aligned}\)

But height cannot be negative thus substitution wrong

Let v=60

substitute 60 for v in the equation

⇒ \(\begin{aligned}
y_0 & =\frac{405000}{(60)^2}-107.5 \\
y_0 & =112.5-107.5 \\
& =5
\end{aligned}\)

Thus the initial height and initial speed of the ball are 5 and 60 respectively.

Thus the initial height and initial speed of the ball are 5 and 60 respectively.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice

Page 8 Exercise 2 Answer

The given expression is 4(xy)2− 2x + 5y.

It is required to find the value of the given expression when the value of x is 6 and y is 2.

To find the value of the given expression, rewrite the given expression. Substitute the given values for x and y. Then solve further to get the result.

The given expression is 4(xy)² – 2x+5y
Substitute 6 for x and 2 for y in the given expression.

4(xy)²-2x+5y = 4(6×2)²-2(6)+5(2)
4(xy)²-2x+5y=4(12)² -2(6)+5(2)
4(xy)²-2x+5y= 4(144)-12+10
4(xy)²-2x+5y=576-2
4(xy)²-2x+5y=574

The value of the expression 4(xy)2 − 2x + 5y is 574.

Page 8 Exercise 3 Answer

It is given that Camilla and Nadia start a business tutoring students in maths. They rent an office for $250 per month and charge $20 per hour per student.

It is required to calculate the profit they make each month if they have 10 students. It is also required to write a linear equation to solve this.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2

To find the profit, subtract the rent amount from the total amount they get from students. Then solve the linear equation and get the result.

Let x be the profit they make each month. Also, there are 4 weeks in a month.

The profit can be calculated by subtracting the rent amount from the total amount they get from students.

So the linear equation is

x=4(10×20)-250​
x=4(200)-250
x=800-250
x=550

Therefore, Camilla and Nadia make a profit of $550 per month.

Camilla and Nadia make a profit of $550 per month. Also the linear equation is x = 4(10×20) − 250.

It is given that Camilla and Nadia start a business tutoring students in maths. They rent an office for $250 per month and charge $20 per hour per student.

If they have 10 students it is required to sketch the graph for the resulting linear equation.

To sketch the graph, use dynamic geometry software.

Let x be the profit they make each month. Also, there are 4 weeks in a month.

The profit can be calculated by subtracting the rent amount from the total amount they get from students.

So the linear equation is x = 4(10×20) − 250.

The graph of the equation is given below.

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice Page 8 Exercise 3 Answer

Hence the graph of the equation x = 4(10×20) − 250 is,

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice Page 8 Exercise 3 Answer

Envision Algebra 1 Chapter 1 Standards Practice 2 Answer Key

Page 9 Exercise 1 Answer

The given sequence is 2,24,46,68,90,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common difference of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common difference is a constant and equals the difference between any two consecutive terms.

Subtract the first two terms

a2 – a1 =24-2
a2 – a1 =22

Subtract the third term from the second term.

a3 – a2 =24-2
a3 – a2 =22

Subtract the Fourth term from the Third term.

a4 – a3 =68-46
a4 – a3 =22

Hence the common difference id d=22

The formula for expressing arithmetic sequences in their recursive form is an = an-1 + d, where d is the common difference of the given sequence, an is the nth element and an-1 is the (n−1)th element.

Substitute 22 for d in the recursive form.

an = an-1 + 22

Hence the correct answer is option A

Option B

The given sequence is 2,24,46,68,90,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common difference of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common difference is a constant and equals the difference between any two consecutive terms.

Subtract the first two terms

a2 – a1 =24-2
a2 – a1 =22

Subtract the third term from the second term.

a3 – a2 =46-24
a3 – a2 =22

Subtract the Fourth term from the Third term.

a4 – a3 =68-46
a4 – a3 =22

Hence the common difference is d=22

The formula for expressing arithmetic sequences in their recursive form is an = an-1 + d, where d is the common difference of the given sequence, an is the nth element and an-1 is the (n−1)th element.

Substitute 22 for d in the recursive form.

an = an-1 + 22

Here a1 is 2 and an is an-1 + 22 which is not equal to an = an-1 − 22, so option B is incorrect.

Option C

The given sequence is 2,24,46,68,90,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common difference of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common difference is a constant and equals the difference between any two consecutive terms.

Subtract the first two terms

a2 – a1 =24-2
a2 – a1 =22

Subtract the third term from the second term.

a3 – a2 =46-24
a3 – a2 =22

Subtract the Fourth term from the Third term.

a4 – a3 =68-46
a4 – a3 =22

Hence the common difference is d=22.

The formula for expressing arithmetic sequences in their recursive form is an = an-1 + d, where d is the common difference of the given sequence, an is the nth element and an-1 is the (n−1)th element.

Substitute 22 for d in the recursive form.

an = an-1 + 22

Here a1 is 2 and an is an-1 + 22 which is not equal to an = an+1 + 22, so option C is incorrect.

Option D

The given sequence is 2,24,46,68,90,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common difference of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common difference is a constant and equals the difference between any two consecutive terms.

Subtract the first two terms

a2 – a1 =24-2
a2 – a1 =22

Subtract the third term from the second term.

a3 – a2 =46-24
a3 – a2 =22

Subtract the Fourth term from the Third term.

a4 – a3 =68-46
a4 – a3 =22

Hence the common difference is d=22

The formula for expressing arithmetic sequences in their recursive form is an = an-1 + d, where d is the common difference of the given sequence, an is the nth element and an-1 is the (n−1)th element.

Substitute 22 for d in the recursive form.

an = an-1 + 22

Here a1 is 2 and an is an-1 + 22 which is not equal to an = an+1 − 22, so option D is incorrect.

The recursive form of the given arithmetic sequence is an = an-1 + 22, so the correct option is A.

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Practice 2 Solutions

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 10 Exercise 1 Answer

Given the formula pV = nRT.

It is asked to solve the given formula for T.

To solve the problem, simplify the given formula pV = nRT, and solve it for T.

P.V=n.R.T

Divide both sides by n.R.

⇒ \(\begin{aligned}
& \frac{P \cdot V}{n \cdot R}=\frac{n \cdot R \cdot T}{n \cdot R} \\
& \frac{P \cdot v}{n \cdot R}=T
\end{aligned}\)

The formula for T is T = \(\frac{p V}{n R}\).

So option C is correct.

Option A

Given the formula pV = nRT.

It is asked to solve the given formula for T.

To solve the problem, simplify the given formula pV = nRT, and solve it for T.

P.V=n.R.T

Divide both sides by n.R.

⇒ \(\begin{aligned}
& \frac{P \cdot V}{n \cdot R}=\frac{n \cdot R \cdot T}{n \cdot R} \\
& \frac{P \cdot v}{n \cdot R}=T
\end{aligned}\)

The formula for is T = \(\frac{p V}{n R}\) which is not equal to,

T = pV – nR

Option B

Given the formula pV = nRT.

It is asked to solve the given formula for T.

To solve the problem, simplify the given formula pV = nRT, and solve it for T.

P.V=n.R.T

Divide both sides by n.R.

⇒ \(\begin{aligned}
& \frac{P \cdot V}{n \cdot R}=\frac{n \cdot R \cdot T}{n \cdot R} \\
& \frac{P \cdot v}{n \cdot R}=T
\end{aligned}\)

The formula for T is T = \(\frac{p V}{n R}\) which is not equal to,

T = pVnR

Option D

Given the formula pV = nRT.

It is asked to solve the given formula for T.

To solve the problem, simplify the given formula pV = nRT, and solve it for.

P.V=n.R.T

Divide both sides by n.R.

⇒ \(\begin{aligned}
& \frac{P \cdot V}{n \cdot R}=\frac{n \cdot R \cdot T}{n \cdot R} \\
& \frac{P \cdot v}{n \cdot R}=T
\end{aligned}\)

The formula for T is T = \(\frac{p V}{n R}\) which is not equal to,

T = pV

The formula for T is T = \(\frac{p \cdot V}{n \cdot R}\), so option C is the correct option.

Page 11 Exercise 1 Answer

Given a system of equations,

4y − 3x = 14

y + 3x = 10

It is asked to solve the given equations.

To solve the problem make any one coefficient of the equation the same and then take a difference of both the equations. It will eliminate one variable, and then solve for x or y. Then put the value of x or y in any of the given equations to get both x and y.

Multiply the second equation by 5.

5(y+3x)=5(10)

5y+15x=50

Subtract the equation (of step 1) from the first given equation.

5y-3x-(5y+15x)=14(50)
5y-3x-5y-15x=-36
-18x=-36
18x=36
x=2

so the value of x is 2

Now put the value of x in the second equation and simplify

y=3×2=10
y+6=10
y=4

So the solved ordered pair is (2,4) so the correct option is A.

Option B

Subtract the equation (of step 1) from the first given equation.

5(y+3x)=5(10)

5y+15x=50

Subtract the equation (of step 1) from the first given equation.

5y-3x-(5y+15x)=14(50)
5y-3x-5y-15x=-36
-18x=-36
18x=36
x=2

so the value of x is 2

Now put the value of x in the second equation and simplify

y=3×2=10
y+6=10
y=4

​So the solved ordered pair is (2,4) which is not equal to (4,2) so option B is not correct.

Option C

Subtract the equation (of step 1) from the first given equation.​

5(y+3x)=5(10)

5y+15x=50

Subtract the equation (of step 1) from the first given equation.

5y-3x-(5y+15x)=14(50)
5y-3x-5y-15x=-36
-18x=-36
18x=36
x=2

so the value of x is 2

Now put the value of x in the second equation and simplify

y=3×2=10
y+6=10
y=4

​So the solved ordered pair is (2,4) which is not equal to (−2,−4) so option C is not correct.

Option D

Subtract the equation (of step 1) from the first given equation.

5(y+3x)=5(10)

5y+15x=50

Subtract the equation (of step 1) from the first given equation.

5y-3x-(5y+15x)=14(50)
5y-3x-5y-15x=-36
-18x=-36
18x=36
x=2

so the value of x is 2

Now put the value of x in the second equation and simplify

y=3×2=10
y+6=10
y=4

​So the solved ordered pair is (2,4) which is not equal to (2,−4) so option D is not correct.

Option A is the correct option that is (2,4).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 11 Exercise 2 Answer

Given a system of equations,

x + 4y = 6

3x − 4y = 14

It is asked to solve the given equations.

To solve the problem add or take the difference of both the equations to eliminate any one variable from the equation, then solve for x or y. Then put the value of x or y in any of the given equations to get both x and y.

Add both of the equations and then simplify.

Given the System of equations,

x+4y=6

3x-4y=14

Add both of the equations and then simplify.

x+4y+(3x-4y) =6+(14)
x+4y+3x-4y=6+14
4x=20
x=5

So the value of x is 5

Now up the value of x in the first equation and simplify

5+4y=6
4y=6-5
4y=1
\(y=\frac{1}{4}\)

So the solved ordered pair is \(\left(5, \frac{1}{4}\right)\).

The ordered pair of the given system of equation is \(\left(5, \frac{1}{4}\right)\).

Page 11 Exercise 3 Answer

Given that the total money spent is $52, the total number of movies watched is 6, the matinee cost is $7 and the evening show costs $12.

It is asked how many of each movie type he attends and to solve it by graphing.

To solve the problem first form the pair of equations according to the given information that is total movie watched and total money spent and then plot the graph using the equation in slope-intercept form that is \(\frac{x}{h}+\frac{y}{k}=1\), where h and k are the intercepts on the axes. The intersection in the graph is the solution of the equations.

Let x be the number of times Alejandro has gone to a matinee and y be the number of times he has attended the evening show. Then, the equation formed by the given data is,

x + y = 6

7x + 12y = 52

Solve this equation and write them in slope-intercept form.

\(\frac{x}{6}+\frac{y}{6}\) = 1

\(\frac{x}{\frac{52}{7}}+\frac{y}{\frac{13}{3}}\) = 1

Plot the lines using intercept on the graph.

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice Page 11 Exercise 3 Answer

Here the intersection point is (4,2).

So x = 4 and y = 2

So he attended 4 matinees and 2 evening shows.

The graphical plot of the equations formed is,

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice Page 11 Exercise 3 Answer

Alejandro attended 4 matinee movies and 2 evening show movies.

It is asked why the intersection of the graphs is the solution.

The intersection of the graph is the solution because it is the only point that satisfies both equations.

The equations are,

The Equations are, x+y=6

7x+12y=52

Multiply the first equation by 7 and subtract the second equation

​(7x+7y)-(7x+12y)=42-52

7x+7y-7x-12y=-10

-5y=-10

\(y=\frac{10}{5}\)

y=2

put the value of y in the first equation

x+y=6
x+2=6

x=6-2
x=4

So the solution is (2,4) which is the same as the intersection point. Also, these are the only points that satisfy both the equations.

The intersection of the graph is the solution because it is the only point that satisfies both equations.

Page 13 Exercise 1 Answer

A system of inequalities is given.

y > −x + 2

y < x + 4

It is asked to identify the graph of the solution of the given system from the options given below.

A

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 1

B

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 2

C

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 3

D

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 4

To solve this question, first consider the given inequalities as simple lines y = −x + 2 and,

y = x + 4. Determine its x and y intercepts for both and draw corresponding lines passing through these points. As the inequality y > −x + 2 is greater than the inequality, shade the area above the line, and for less than inequality y < x + 4, shade the area below the line. The common shaded area will be the solution for the given system.

Substitute 0 for y and determine the x-intercept of the line y = −x + 2.

0 = −x + 2

−x = −2

x = 2

Thus the x-intercept of the line y = −x + 2 is (2,0).

Substitute 0 for x and determine the y intercept of the line y = −x + 2.

y = 0 + 2

y = 2

Thus the y-intercept of the line y = −x + 2 is (0,2).

Substitute 0 for y and determine the x intercept of the line y = x + 4.

0 = x + 4

x = −4

Thus the x-intercept of the line y = x + 4 is (−4,0).

Substitute 0 for x and determine the y intercept of the line y = x + 4.

y = 0 + 4

y = 4

Thus the y-intercept of the line y = x + 4 is (0,4).

Plot the points (2,0) and (0,2), draw a line passing through them, and shade the area above the line to show the inequality y > −x + 2. Similarly, Plot the points (−4,0) and (0,4), draw a line passing through them, and shade the area below the line to show the inequality y < x + 4.

The common shaded area is the solution of the system.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 5

Compare the graphs given in the options and determine the correct answer.

The graph given in option A is exactly similar to the obtained graph which represents the solution of the system of inequalities,

y > −x + 2

y < x + 4

The graph given in option B represents the system of inequality,

y > x + 2

y > x + 4

The graph given in option C represents the system of inequality,

y < x + 2

y < x + 4

The graph given in option D represents the system of inequality,

y < −x + 2

y < x + 4

Thus, option A is the correct answer.

The graph of the solution of the given system is obtained and therefore, option A is the correct answer.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 1 Answer Image 6

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 13 Exercise 2 Answer

The function g(x) = \(\frac{1}{3}|x-2|+10\) is given.

It is asked to compare the given graph with the parent graph f(x) = ∣x∣.

To solve this question, first plot the graphs of f(x) = ∣x∣ and,

g(x) = \(\frac{1}{3}|x-2|+10\). Then observe the second graph and how it changes concerning the first graph.

Ploth the graphs of f(x) = ∣x∣ and,g(x) = \(\frac{1}{3}|x-2|+10\) in a single screen and compare them.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 2 Answer

In the function, the value in the mode is x−2 which means that it is shifted 2 units to the right side of the parent graph which can be seen from the graph.

Thus, \(\frac{1}{3}\) is multiplied with |x – 2|, which is greater than zero and less than one, which indicates that the graph is vertically stretched by \(\frac{1}{3}\) units from the parent graph.

Finally, 10 is added to \(\frac{1}{3}\)|x-2|, which indicates that the graph is shifted 10 units vertically upward which also can be seen from the graph.

The graph of g(x) = \(\frac{1}{3}|x-2|+10\) is compared with its parent graph,

f(x) = |x| and three important results are obtained.

First, horizontal shift of 2 units to the right side.

Second, vertical stretch of \(\frac{1}{3}\)|x-2| units.

Third, a Vertical shift of 10 units in the upward direction.

Envision Algebra 1 Student Edition Standards Practice 2 Guide

Page 13 Exercise 3 Answer

It is given that Manuel wants to raise between $250 and $350 for charity. His parents donated $70 and he plans to ask others to contribute $10.

It is asked to write an inequality to determine the number of people who will be needed to contribute for Manuel to reach his goal.

To obtain the inequality, let the number of contributors be x. Each contributor will need to donate $10.Therefore, the total amount to be collected is $10x.

But it is given that Manuel already has $70 donated by his parents. Let y be the total amount of charity he would collect, then y = 70 + 10x.

This amount will be between $250 and $350. Therefore, the inequality to solve the problem will be,
Given, 250<y<350

250<70+10x<35

Simplify Further,

250-70<10x<350-70
180<10x<280
18<x<28

The inequality to solve the problem is 18 < x < 28.

It is given that Manuel wants to raise between $250 and $350 for charity. His parents donated $70 and he plans to ask others to contribute $10.

It is asked to show the solution on the graph to determine the number of people who will be needed to contribute for Manuel to reach his goal and explain the same in words.

To solve this question, use the inequality obtained in the previous part of this exercise which is,

250 < 70 + 10x < 350. First, solve this inequality for x and then graph the solution.

Solve the inequality 250 < 70 + 10x < 350 for x.

Subtract 70 forms from all parts of inequality,

250 − 70 < 70 + 10x − 70 < 350 − 70

180 < 10x < 280

Divide by 10 in all parts of inequality,

⇒\(\frac{180}{10}<\frac{10 x}{10}<\frac{280}{10}\)

18 < x < 28

Thus, the number of people who will be needed to contribute for Manuel to reach his goal is between 18 and 28.

To graph this solution, first plot the line y = 70 + 10x.

After that show the range of the money collection which is 250 < y < 350. For this, draw two lines, y = 250 and, y = 350, and shade the area between them.

After that from the intersection points of this region and the line y = 70 + 10x, draw two vertical lines that will give the solution of the system which is the range 18 < x < 28.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 3 Answer

The number of people who will be needed to contribute for Manuel to reach his goal is between 18 and 28. The graph of the solution is obtained.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 13 Exercise 3 Answer

 

Page 14 Exercise 1 Answer

An equation is given as \(\frac{x}{3}=\frac{10}{15}\).

It is required to solve the given equation for x.

Given equation \(\frac{x}{3}=\frac{10}{15}\)

Multiply both sides of the equation \(\frac{x}{3}=\frac{10}{15} \text { by } 3 .\)

Hence, option A is correct.

Option B

Multiply both sides of the equation,

Given equation \(\frac{x}{3}=\frac{10}{15} .\)

Multiply both sides of the equation \(\frac{x}{3}=\frac{10}{15}\) by 3

\(\begin{aligned}
& \frac{x}{3} \times 3=\frac{10}{15} \times 3 \\
& x=\frac{30}{15} \\
& x=2
\end{aligned}\)

In Option B, the value of x is given as 3.

Hence, option B is incorrect.

Option C

Multiply both sides of the equation,

Given equation \(
\frac{x}{3}=\frac{10}{15}\) 

Multiple both sides of the equation \(\frac{x}{3}=\frac{10}{15} \text { by } 3 \text {. }\)

\(\begin{aligned}
& \frac{x}{3} \times 3=\frac{10}{15} \times 3 \\
& x=\frac{30}{15} \\
& x=2
\end{aligned}\)

In option C, the value of x is given as 12. Hence, option C is incorrect.

Option D

Multiply both sides of the equation,

Given equation \(\frac{x}{3}=\frac{10}{15}\) 

Multiple both sides of the equation \(\frac{x}{3}=\frac{10}{15} \text { by } 3 \text {. }\)

\(\begin{aligned}
& \frac{x}{3} \times 3=\frac{10}{15} \times 3 \\
& x=\frac{30}{15} \\
& x=2
\end{aligned}\)

In option D, the value of x is given as 15. Hence, option D is incorrect.

The correct answer is option A, the value of x is 2.

Page 14 Exercise 2 Answer

A function is given as,

⇒\(f(x)=\left\{\begin{array}{l}
1-0.5 x, x<1 \\
x, x \geq 1
\end{array}\right.\)

It is required to graph the given function.

To do so, draw the given lines, and then for the line y = 1 − 0.5x, limit the line for the values where x is greater than 1. For the line y = x, limit the line for values where x is smaller than or equal to 1.

Draw the lines on the graph,

y = 1 − 0.5x

y = x
Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 14 Exercise 2 Answer Image 1

Limit the line for the values where x is greater than 1. For the line y = x, limit the line for values where x is smaller than or equal to 1.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 14 Exercise 2 Answer Image 2

The graph of the given function is drawn as,

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 14 Exercise 2 Answer Image 2

 

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 14 Exercise 3 Answer

It is given that the length of a rectangle is 4 more than twice its width and the perimeter can be no more than 92 ft.

It is required to write an inequality to find all the possible values of the width of the rectangle.

To do so, let the width be x. Form the expression for the length and perimeter of a rectangle. Apply the given condition that the perimeter can be no more than 92 ft.

Let the width be x.

Given that the length of the rectangle is 4 more than twice its width. Hence, the length will be 2x+4.

The ​Perimeter of the rectangle, with x width and 2x+4 length, is

p=2((x)+(2x+4))

p=2(3x+4)

It is given that the perimeter can be no more than 92ft. Hence,

2(3x+4)≤92
3x+4≤46
3x≤42
x≤14

The width cannot be negative and 0. Hence, the inequality for width x is, 0 < x ≤ 14.

The inequality to solve the problem is 0 < x ≤ 14.

It is given that the length of a rectangle is 4 more than twice its width and the perimeter can be no more than 92 ft.

It is required to graph the inequality and describe the solution in words.

To do so, draw a graph only with the x-axis and represent the inequality on the graph. Also, describe the solution in words.

The inequality for width x is, 0 < x ≤ 14.

Represent the inequality on the graph.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 14 Exercise 3 Answer

The possible values of width are greater than 0 and less than or equal to 14 ft.

The representation of inequality as a graph is,

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 14 Exercise 3 Answer

The solution in words is written as the possible value of width is greater than 0 and less than or equal to 14 ft.

 

Page 15 Exercise 1 Answer

The given expression is 6−2x3y−5.

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

Given Expression 6-3 x3 y-5

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{6^2 y^5}\)

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{36 y^5}\)

So the correct option is C.

Option A

The given expression is 6−2x3y−5.

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

Given Expression 6-3 x3 y-5

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{6^2 y^5}\)

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{36 y^5}\)

The simplified form is \(\frac{x^3}{36 y^5}\) which is not equal to -12x35y so option A is incorrect.

Option B

The given expression is 6−2x3y−5.

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

Given Expression 6-3 x3 y-5

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{6^2 y^5}\)

⇒ \(6^{-2} x^3 y^{-5}=\frac{x^3}{36 y^5}\)

The simplified form is \(\frac{x^3}{36 y^5}\) which is not equal to -36x3y so option B is incorrect.

Option D

The given expression is 6−2x3y−5.

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

6−2x3y−5 = \(\frac{x^3}{6^2 y^5}\)

6−2x3y−5The simplified form is \(\frac{x^3}{36 y^5}\) which is not equal to \(\frac{x^3}{12 y^5}\) so the option D is incorrect.

The simplified form of 6−2x3y−5 is \(\frac{x^3}{36 y^5}\), so the correct option is C.

Page 15 Exercise 2 Answer

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\).

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5 a t^4 \times 25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5^3 a^3 t^4}\)

⇒ \(\begin{aligned}
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5^{\frac{3}{3}} a^{\frac{3}{3}} t^{\frac{4}{3}} \\
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5 a t^{4 / 3}
\end{aligned}\)

So the correct option is C.

Option A

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\).

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5 a t^4 \times 25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5^3 a^3 t^4}\)

⇒ \(\begin{aligned}
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5^{\frac{3}{3}} a^{\frac{3}{3}} t^{\frac{4}{3}} \\
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5 a t^{4 / 3}
\end{aligned}\)

The simplifed form is \(5 a t^{\frac{4}{3}}\) which is not equal to \(5^{\frac{1}{3}} a t^2\) so the option A is incorrect.

Option B

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\).

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5 a t^4 \times 25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5^3 a^3 t^4}\)

⇒ \(\begin{aligned}
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5^{\frac{3}{3}} a^{\frac{3}{3}} t^{\frac{4}{3}} \\
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5 a t^{4 / 3}
\end{aligned}\)

The simplifed form is \(5 a t^{\frac{4}{3}}\) which is not equal to \(5^{\frac{2}{3}} a^{\frac{1}{3}} t^{\frac{4}{3}}\) so the option B is incorrect.

Option D

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\).

It is required to simplify the given expression.

To simplify the given expression use the algebraic identity.

The given expression is \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5 a t^4 \times 25 a^2}\)

⇒ \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=\sqrt[3]{5^3 a^3 t^4}\)

⇒ \(\begin{aligned}
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5^{\frac{3}{3}} a^{\frac{3}{3}} t^{\frac{4}{3}} \\
& \sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}=5 a t^{4 / 3}
\end{aligned}\)

The simplified form is \(5 a t^{\frac{4}{3}}\) which is not equal to 125a3t4 so option D is incorrect.

The simplified form of \(\sqrt[3]{5 a t^4} \times \sqrt[3]{25 a^2}\) is \(5 a t^{\frac{4}{3}}\), so the correct option is C.

How To Solve Envision Algebra 1 Standards Practice 2 Problems

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 15 Exercise 3 Answer

The given expression is \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{5}}\right)\).

It is required to simplify the given expression.

To simplify the given expression, use the algebraic identity.

Rewrite the given expression; use the algebraic identity aman = am+n to simplify the expression.

The Given expression is \(\left(4g \frac{1}{3} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{5}}\right)\)

⇒ \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{2}}\right)=(4 \cdot 2 \cdot 3)\left(g^{\frac{1}{3}} \cdot g^{\frac{2}{3}}\right)\left(h^{\frac{3}{5}} \cdot h^{\frac{1}{5}}\right)\)

⇒ \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{2}}\right)=(24)\left(9 \frac{1}{3}+\frac{2}{3}\right)\left(h^{\frac{3}{5}+\frac{1}{5}}\right)\)

⇒ \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{2}}\right)=(24)\left(g^{\frac{3}{3}}\right)\left(h^{\frac{4}{5}}\right)\)

⇒ \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{2}}\right)=24 g h^{\frac{4}{5}}\)

The simplified form of the given expression \(\left(4 g^{\frac{1}{3}} 2 h^{\frac{3}{5}}\right)\left(3 g^{\frac{2}{3}} h^{\frac{1}{5}}\right)\) is \(24 g h^{\frac{4}{5}}\).

Page 15 Exercise 4 Answer

It is given that b−(xy) can be written as \(\frac{1}{\left(b^x\right)^y}\) for all real numbers x and y.

It is required to find whether the given statement is correct and show examples to justify your explanation.

To simplify the given expression, use the algebraic identity.

Use the algebraic identity to simplify the given expression.

b-(xy) = \(\frac{1}{b^{(x y)}}\)

b-(xy) = \(\frac{1}{\left(b^x\right)^y}\)

Hence the given statement is correct.

The given statement b-(xy) = \(\frac{1}{\left(b^x\right)^y}\) is correct.

Page 16 Exercise 1 Answer

The given sequence is 4,16,64,256,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common ratio of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common ratio is a constant and equals the ratio between any two consecutive terms.

Divide the first two terms

⇒ \(\begin{aligned}
& \frac{a_2}{a_1}=\frac{16}{4} \\
& \frac{a_2}{a_1}=4
\end{aligned}\)

Divide the third term and the second term

⇒ \(\begin{aligned}
& \frac{a_3}{a_2}=\frac{64}{16} \\
& \frac{a_3}{a_2}=4
\end{aligned}\)

Hence the common ratio is r=4

The formula for expressing geometric sequences in their recursive form is an = an−1⋅r, where r is the common ratio of the given sequence, an is the nth element and an−1 is the (n−1)th element.

Substitute 4 for r in the recursive form.

an = an−1 ⋅ 4

an = 4an−1

​Hence the correct answer is option B.

Option A

The given sequence is 4,16,64,256,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common ratio of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common ratio is a constant and equals the ratio between any two consecutive terms.

Divide the first two terms

⇒ \(\begin{aligned}
& \frac{a_2}{a_1}=\frac{16}{4} \\
& \frac{a_2}{a_1}=4
\end{aligned}\)

Divide the third term and the second term

⇒ \(\begin{aligned}
& \frac{a_3}{a_2}=\frac{64}{16} \\
& \frac{a_3}{a_2}=4
\end{aligned}\)

Hence the common ratio is r=4

The formula for expressing geometric sequences in their recursive form is an = an−1⋅r, where r is the common ratio of the given sequence, an is the nth element and an−1 is the (n−1)th element.

Substitute 4 for r in the recursive form.

an = an−1 ⋅ 4

an = 4an−1

Here a1 is 4 and an is 4an−1 which is not equal to a1 = 0, so option A is incorrect.

Option C

The given sequence is 4,16,64,256,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common ratio of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common ratio is a constant and equals the ratio between any two consecutive terms.

Divide the first two terms

⇒ \(\begin{aligned}
& \frac{a_2}{a_1}=\frac{16}{4} \\
& \frac{a_2}{a_1}=4
\end{aligned}\)

Divide the third term and the second term

⇒ \(\begin{aligned}
& \frac{a_3}{a_2}=\frac{64}{16} \\
& \frac{a_3}{a_2}=4
\end{aligned}\)

Hence the common ratio is r=4

The formula for expressing geometric sequences in their recursive form is an = an−1⋅r, where r is the common ratio of the given sequence, an is the nth element and an−1 is the (n−1)th element.

Substitute 4 for r in the recursive form.

an = an−1 ⋅ 4

an = 4an−1

Here a1 is 4 and an is 4an−1  which is not equal to an = 4 + 4an−1, so option C is incorrect.

Option D

The given sequence is 4,16,64,256,…

It is required to find the recursive formula for the given sequence.

To find the recursive formula, find the common ratio of the given sequence and substitute it in the formula for expressing arithmetic sequences in their recursive form.

The common ratio is a constant and equals the ratio between any two consecutive terms.

Divide the first two terms

⇒ \(\begin{aligned}
& \frac{a_2}{a_1}=\frac{16}{4} \\
& \frac{a_2}{a_1}=4
\end{aligned}\)

Divide the third term and the second term

⇒ \(\begin{aligned}
& \frac{a_3}{a_2}=\frac{64}{16} \\
& \frac{a_3}{a_2}=4
\end{aligned}\)

Hence the common ratio is r=4

The formula for expressing geometric sequences in their recursive form is an = an−1⋅r, where r is the common ratio of the given sequence, an is the nth element and an−1 is the (n−1)th element.

Substitute 4 for r in the recursive form.

an = an−1 ⋅ 4

an = 4an−1

Here a1 is 4 an is 4an−1 which is not equal to a1 = 0 and an is 4 + an−1, so option D is incorrect.

Hence the correct answer is option B.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 17 Exercise 1 Answer

Given the expression 5n6 − n4 − 3n2 − 2n + 4.

It is required to find the number of terms in the expression.

The number of terms is usually the number of terms in the fully expanded form of the polynomial.

In the given expression there are 5 terms and the degree of every term is different, so the expression cannot be simplified further.

Hence the number of terms in the given expression is 5.

The correct option is C.

The given expression is 5n6 − n4 − 3n2 − 2n + 4.

Since the degrees in the expression are different and the expression is in fully expanded form. So, the number of terms in the expression is exactly 5, not more or less. So, the options A, B, and D are not correct.

The correct answer is Option (C), the number of terms in the given expression is 5.

Page 17 Exercise 2 Answer

Given the monomial 4a5bc.

It is required to find the number of terms in the monomial. The degree of a monomial is the sum of the exponents of the variable.

The sum of the exponents of the variable is 5 + 1 + 1 = 7.

Hence the degree of the monomial is 7.

The correct option is D.

Given the monomial 4a5bc.

The degree of a monomial is the sum of the exponents of the variable.

The sum of the exponents of the variable is 7.

Hence options A, B, and C are incorrect.

The correct answer is Option (D), the degree of the monomial is 7.

Page 17 Exercise 4 Answer

A new house worth $250,000 depreciates at a rate of 16% a year. It is required to explain the situation in terms of growth or decay. If the value increases with time, then the given situation is of growth and if the value decreases with time, then the given situation is of decay. In the given situation, the value of the house decreases by 16% every year. Therefore the given situation is of decay.

The given situation is of decay because the value of the house reduces every year by 16%.

A new house worth $250,000 depreciates at a rate of 16% a year.

It is required to write a function to model the situation.

To do so, find the worth of the house after one year and after two years. Observe the sequence of the initial worth of the house, worth after one year, worth after two years, and as a result, a function can be written to model the situation.

Find the worth of the house after one year.

The initial worth of the house is $250,000.

The worth of the house after one year will be,

Worth after one year = 250,000 − 0.16 × 250,000

Worth after one year = 250,000 × 0.84

Find the worth of the house after two years.

The worth of the house after two years will be,

Worth after two year = 250,000 − 0.16 × (250,000×0.84)

Worth after two year = 250,000 × 0.84 × 0.84

Model the function of the situation.

Let W be the worth of the house after n years.

The function of this situation can be modeled as,

W = $250,000 × (0.84)n

The function to model the situation can be represented as W = $250,000 × (0.84)n, where W is the worth of the house and n is the number of years.

A new house worth $250,000 depreciates at a rate of 16% a year.

It is required to write a function to model the situation.

To do so, find the worth of the house after one year and after two years. Observe the sequence of the initial worth of the house, worth after one year, worth after two years, and as a result, a function can be written to model the situation. As a result, the value of the house after 5 years can be determined.

Find the worth of the house after one year.

The initial worth of the house is $250,000.

The worth of the house after one year will be,

Worth after one year = 250,000 − 0.16 × 250,000

Worth after one year = 250,000 × 0.84

Find the worth of the house after two years.

The worth of the house after two years will be,

Worth after two year = 250,000 − 0.16 × (250,000×0.84)

Worth after two year = 250,000 × 0.84 × 0.84

Find the function for the situation.

Let W be the worth of the house after n years.

The function of this situation can be modeled as

w=$250000x(0.84)n

Find the value of the house after 5 years

substitute 5 for n in the function

w=$ 250,000 x(0.84)n

w=$ 250,000 x(0.84)5

w=$ 250,000 x(0.4182

w=$104,550

The worth of the house after 5 years is $104,550.

Envision Algebra 1 Standards Practice 2 Step-By-Step Explanation

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 18 Exercise 1 Answer

The given expression is (x−3)2.

It is required to solve the given expression.

Use the algebraic identity (a−b)2 = a2 − 2ab + b2 to simplify the given expression. Choose x for a and 3 for b.

​Given Expression, (x-3)²

(a-b)² =a² -2ab+b²
(x-3)²=x²-2(x)(3)+(3)²
(x-3)²=x²-6x+9

So the correct option is C.

Option A

The given expression is (x−3)2.

It is required to solve the given expression.

Use the algebraic identity (a−b)2 = a2 − 2ab + b2 to simplify the given expression. Choose x for a and 3 for b.

​Given Expression, (x-3)²

(a-b)² =a² -2ab+b²
(x-3)²=x²-2(x)(3)+(3)²
(x-3)²=x²-6x+9

The simplified form of the given expression is x2 – 6x + 9 which is not equal to 2x – 6 so option A is incorrect.

Option B

The given expression is (x−3)2.

It is required to solve the given expression.

Use the algebraic identity (a−b)2 = a2 − 2ab + b2 to simplify the given expression. Choose x for a and 3 for b.

​Given Expression, (x-3)²

(a-b)² =a² -2ab+b²
(x-3)²=x²-2(x)(3)+(3)²
(x-3)²=x²-6x+9​

The simplified form of the given expression is x2 – 6x + 9 which is not equal to x2 + 9 so option B is incorrect.

Option D

The given expression is (x−3)2.

It is required to solve the given expression.

Use the algebraic identity (a−b)2 = a2 − 2ab + b2 to simplify the given expression. Choose x for a and 3 for b.

​Given Expression, (x-3)²

(a-b)² =a² -2ab+b²
(x-3)²=x²-2(x)(3)+(3)²
(x-3)²=x²-6x+9​

The simplified form of the given expression is x2 – 6x + 9 which is not equal to x2 – 9x – 6 so option D is incorrect.

The simplified form of x2 − 6x + 9 is x2 − 6x + 9, so the correct option is C.

Page 18 Exercise 2 Answer

The given expressions are,

(5x2 + 10x + 6) − (2x2 − 4x + 6)

x(3x + 14)

6x2 − (9x2 + 12x) − 2x

(x2 + 6x2) + (−4x2 + 2x − 4x + 16x)

It is required to classify the given equations as those equivalent to 3x2 + 14x and not equivalent to 3x2 + 14x.

To do so, simplify each given expression and check whether it is equivalent to 3x2 + 14x or not.

Simplify the first given expression (5x2 + 10x + 6) − (2x2 − 4x + 6).

Given expression (5x²+10x+6)-(2x²-4x+6)
(5x²+10x+6)-(2x²-4x+6)= 5x²+10x+6-2x²+4x-6
(5x²+10x+6)-(2x²-4x+6)= (5-2)x² +(10+4)x+6-6
(5x²+10x+6)-2x²-4c+6)=3x²+14

Therefore, (5x2 + 10x + 6) − (2x2 − 4x + 6) is equivalent to 3x2 + 14x.

Simplify the second given expression x (3x + 14).

x(3x + 14) = 3x2 + 14x

Therefore, x(3x+14) is equivalent to 3x2 + 14x.

Simplify the third given expression 6x2 − (9x2 + 12x) − 2x.

Given expression ​6x²-(9x² +12x)-2x

6x²-(9x²+12x)-2x=6x²-9x²-12x-2x
6x²-(9x²+12x)-2x=(6-9)x²-(12-2)x
6x²-(9x²+12x)-2x=-3x²-14x

Therefore, 6x2 − (9x2 + 12x) − 2x is not equivalent to 3x2 + 14x.

Simplify the fourth given expression (x2 + 6x2) + (−4x2 + 2x − 4x + 16x).

Given Expression (x²+6x²)+(-4x²+2x-4x+16x)
(x²+6x²)+(-4x²+2x-4x+16x)=x²+6x²-4x²+2x-4x+16)x
(x²+6x²)+(-4x²+2x-4x+16x)=(1+6-4)x²+(2-4+16)x
(x²+6x²)+ (-4x²+2x-4x+16x)=3x²+14x

Therefore, (x2 + 6x2) + (−4x2 + 2x −4x + 16x) is equivalent to 3x2 + 14x.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 18 Exercise 2 Answer

Page 18 Exercise 3 Answer

It is given that the length of a rectangular sandbox is 4x + 1 and the width of the sandbox is x − 2.

It is required to find a polynomial in standard form which represents the area of the sandbox.

To do so, use the area of the rectangle formula. Substitute the given values of length and breadth. Solve further to find the polynomial.

The area of a rectangle is A=lw.

Substitute 4x+1 for 1 and x-2 for w

Solve further to find the formula

A=(4x+1)(x-2)
A=(4x)(x)-2(4x)+1(x)+1(-2)
A=4x²-8x+x-2
A=4x²-7x-2

Hence the polynomial in standard form which represents the area of the sandbox is 4x2 − 7x − 2.

It is given that the length of a rectangular sandbox is 4x+1 and the width of the sandbox is x−2.

It is required to find a polynomial in standard form which represents the area of the sandbox. And name the polynomial based on its degree and number of terms.

To do so, use the area of the rectangle formula. Substitute the given values of length and breadth. Solve further to find the polynomial. Then name the polynomial based on its degree and number of terms.

The area of a rectangle is A=lw.

Substitute 4x+1 for 1 and x-2 for w

Solve further to find the formula

A=(4x+1)(x-2)
A=(4x)(x)-2(4x)+1(x)+1(-2)
A=4x²-8x+x-2
A=4x²-7x-2​

The polynomial is 4x2 − 7x − 2.

The name of the polynomial 4x2 − 7x − 2 based on the degree is a quadratic polynomial because this polynomial has degree 2.

The name of the polynomial 4x2 − 7x − 2 based on the number of terms is trinomial because this polynomial has three terms.

The name of the polynomial 4x2 − 7x − 2 based on the number of terms is trinomial and its name based on the degree is quadratic polynomial.

Page 19 Exercise 1 Answer

The given polynomial is 5x2 + 27x − 18.

It is required to factorize the given polynomial.

The Given polynomial is 5x²+27x-18

5x²+27x-18=5x²+30x-3x-18
5x²+27x-18=5x(x+6)-3(x+6)
5x²+27x-18=(x+6)(5x-3)

​So the correct answer is option C.

Option A

Factorize the given polynomial 5x2 + 27x − 18.

The Given polynomial is 5x²+27x-18

5x²+27x-18=5x²+30x-3x-18
5x²+27x-18=5x(x+6)-3(x+6)
5x²+27x-18=(x+6)(5x-3)

Which is not equivalent to (5x+3)(x+6)

so option A is incorrect.

Option B

Factorize the given polynomial 5x2 + 27x − 18.

The Given polynomial is 5x²+27x-18

5x²+27x-18=5x²+30x-3x-18
5x²+27x-18=5x(x+6)-3(x+6)
5x²+27x-18=(x+6)(5x-3)

Which is not equivalent to (5x−2)(x−9)

so option B is incorrect.

Option D

Factorize the given polynomial 5x2 + 27x − 18.

The Given polynomial is 5x²+27x-18

5x²+27x-18=5x²+30x-3x-18
5x²+27x-18=5x(x+6)-3(x+6)
5x²+27x-18=(x+6)(5x-3)

Which is not equivalent to (5x+2)(x−9)

so option D is incorrect.

The factor of the polynomial 5x2 + 27x − 18 is (5x−3)(x+6), so the correct option is C.

Page 19 Exercise 2 Answer

The given expression is (x−3)2.

It is required to simplify the given expression.

To solve the given expression, use the algebraic identity.

Use the algebraic identity (a−b)3 = a3 − 3a2b + 3ab2 − b3. Choose x for a and 3 for b and solve further.

The given expression, (x-3)³
(a-b)³=a³-3a²b+3ab²-b³
(x-3)³=x³-9x²+3x(3)²-(3)³

​The simplified form of (x−3)3 is x3 − 9x2 + 27x − 27.

The given expression is (x−3)2.

It is required to simplify the given expression and name it based on the degree.

To solve the given expression, use the algebraic identity. Name the simplified form based on degree.

Use the algebraic identity (a−b)3 = a3 − 3a2b + 3ab2 − b3. Choose x for a and 3 for b and solve further.

The given expression,(x-3)²

(a-b)³=a³-3a²b+3ab²-b³
(x-3)³=x³-9x²+3x(3)²-(3)³
(x-3)³=x³-9x²+27x-27

This is a cubic polynomial because it has degree 3.

The name of the polynomial x3 − 9x2 + 27x − 27 based on degree is a cubic polynomial.

Page 19 Exercise 3 Answer

A figure is given in the question.

It is required to find the area of the rectangle given in the figure.

To do so, use the area of the rectangle formula. Substitute the given values of length and breadth. Solve further to find the polynomial.

The given figure is,

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 19 Exercise 3 Answer Image 1

From the figure, the length of the rectangle is 2x + 8 and its width is x + 4.

The area Of a rectangle is A=lw

Substitute 2x+8 for 1 and x+4 for w

solve further to find the polynomial

A=(2x+8)(x+4)
A=(2x)(x)+2x)(4)+8(x)+8(4)
A=2x²+8x+8x+32
A=2x²+16x+32

The area of the rectangle in the given figure is 2x2 + 16x + 32.

A figure is given in the question. It is also given that the value of π is 3.14.

It is required to find the area of the circle given in the figure.

To do so, use the area of the circle formula. Substitute the given values of radius. Solve further to find the polynomial.

The given figure is,

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 19 Exercise 3 Answer Image 2

From the figure, the radius of the circle is x+4.

The area of a circle is A= πr²

Substitute x+4 for r and 3.14 for π

Solve further to find the polynomial

A=(3.14)(x+4)²
A=(3.14)(x²+8x+16
A=(3.14)(x²)+(3.14)(8x)+(3.14)(16)
A=3.14×2+25.12x+50.24

The area of the circle in the given figure is 3.14x2 + 25.12x + 50.24.

A figure is given in the question. It is also given that the value of π is 3.14.

It is required to find the area of the shaded region given in the figure.

To do so, find the area of the circle and the area of the rectangle. Then subtract the area of the circle from the area of the rectangle.

The given figure is,

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 19 Exercise 3 Answer Image 3

From the figure, the radius of the circle is x+4.

The area of a circle is A= πr²

Substitute x+4 for r and 3.14 for π

Solve further to find the polynomial

A1=(3.14)(x+4)²
A1=(3.14)(x²+8x+16)
A1=(3.14)x²+(3.14)(8x)+(3.14)(16)
A1=3.14x²+25.12x+50.24

From the figure, the length of the rectangle is 2x+8 and its width is x+4.

The area of a rectangle is A=lw

Substitute 2x+8 for 1 and x+4=w

Solve further to find the polynomial.

A2=(2x+8)(x+4)
A2=(2x)(x)+(2x)(4)+8(x)+8(4)
A2=2x²+8x+8x+32
A2=2x²+16x+32

To find The area of the shaded region,

subtract the area of the circle from the area of the rectangle

A=(2x²+16x+32)-)(3.14x²+25.12x+50.24
A=(2-3.14)x²+(16-25.12)x+32-50.24
A=-1.14×2-9.12x-18.24

The area of the shaded region is 3.14x2 + 25.12x + 50.24.

Page 20 Exercise 2 Answer

Given is the polynomial representing the area of a square.

64x2 + 96x + 36

It is asked to find the expression for the length of the sides.

To solve the problem use the formula for the area of the square that is side2 and equate it to the given polynomial and then simplify and solve for the side.

Equate the given polynomial with the area of the square that is side 2.

Given Expression 64x² +96x+36

Side² =4((4x)²+2.4x.3+3²)
Side²=(2(4x+3))²
Side2=8x+6

So the expression for the side of the square is (8x+6).

The expression for the side length is (8x+6) units.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 2 Page 20 Exercise 3 Answer

Given, the dimension of the garden is 12m × 8m. A pathway of xm is to be made around the perimeter of the garden having the same area as the garden.

It is asked to sketch the situation and to find a polynomial representing the total area of the garden and pathway.

To solve the problem first sketch the situation and then find the area of the garden using the formula for the area of the rectangle substituting the values of length and breadth

Then find the area of the pathway assuming it to be made up of four different rectangles.

Then add the area of the garden and the area of the pathway to get the polynomial representing the total area of the garden and the pathway.

Sketch of the given situation.

Envision Algebra 1 Assessment Chapter 1 Standards Practice Page 20 Exercise 3 Answer

The length of the garden is l = 12m.
Breadth of the garden is b=8m
The area of the garden is,

A=lxb
A=12×8
A=96

So the area of the garden is 96m2.

Assume that the pathway is formed of 4 rectangles. Two rectangles having the length of 12m and breadth of xm, and other two rectangles having the length of 8m + 2xm and breadth of xm, then the total area of the pathway is given by,

A=lxb+lxb+lxb+lxb

⇒ \(\begin{aligned}
& A=(12 \times x)+(12 \times x)+((8+2 x) \times x)+((8+2 x) \times x) \\
& A=2 \cdot(12 \times x)+2 \cdot((8+2 x) \times x) \\
& A=24 x+16 x+4 x^2 \\
& A=4 x^2+40 x
\end{aligned}\)

​So the area of the pathway is A = 4x2 + 40x.

The total area will be the sum of the area of the pathway and the area of the garden.

Total Area= Area of the garden+Area of the pathway

2A=96+4x²+40x
A=48+2x²+20x
A=2x²+20x+48

The polynomial for the total area of the garden and the pathway is 2x2 + 20x + 48.

Given, the dimension of the garden is 12m × 8m. A pathway of xm is to be made around the perimeter of the garden having the same area as the garden.

It is asked to find the width of the pathway.

To solve the problem, first equate the area of the pathway from the previous problem with the area of the garden to get a polynomial. Then factorise the polynomial and simplify and solve for x.

The area of the garden is 96m2 and the area of the pathway from the previous part of the question is A = 4×2 + 40x.

Equate the area of the pathway with the area of the garden.

\(\begin{aligned}
& 4 x^2+40 x=96 \\
& x^2+10 x=24 \\
& x^2+10 x-24=0
\end{aligned}\)

So the polynomial is x2+10x-24=0

Now factorize the polynomial to get the value of x.

\(\begin{aligned}
& x^2+10 x-24=0 \\
& x^2+12 x-2 x-24=0 \\
& x(x+12)-2(x+12)=0 \\
& (x-2) \cdot(x+12)=0
\end{aligned}\)

So x = 2 or −12 but as length cannot be negative,

x = 2m

As x is the width of the pathway so width of the pathway is 2m.

The width of the pathway is 2 meters.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1

Envision Algebra 1 Assessment Readiness Workbook Chapter 1 Standards Practice

Page 2 Exercise 1 Answer

It is given that the expression 0.25b + 6 models the total cost to hit b baseball in the batting cages.

It is asked to determine the cost per baseball from the given options.

A. $6.25

B. $6.00

C. $0.25

D. $0.19

The cost per baseball means the cost to hit a single baseball. This means the required answer can be obtained by substituting the value 1 for b in the given expression.

Substitute the value 1 for b in the expression 0.25b + 6 to get the cost of hitting 1 baseball. ​

Given 0.25b +6
b=1
Substitute the value of b
0.25b+6
=0.25(1)+6
=0.25(1)+6
=0.25+6
=6.25

Thus, the cost per baseball is $6.25.

The value of the expression 0.25b + 6 can be $6.00 only when no baseball is hit and here it is asked to find the cost to hit a single baseball.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1

Therefore, option B is the wrong answer.

Also, the minimum value of the expression 0.25b + 6 will be $6.00 even if no baseball is hit, and the values given in options C and D are less than $6.00 which is not possible.

Therefore, options C and D also have wrong answers.

The cost per baseball is $6.25 and therefore, option A is the right answer.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 2 Exercise 2 Answer

The following figure is given.

Standards Practice Page 2 Exercise 2 Answer

It is asked to find the measure of each angle.

The sum of all the angles of the triangle is 180°. Apply this rule on △ABC, substitute the given values, and then solve the equation to determine the value of the variable x. Then substitute the obtained value in each angle and find their measures.

Apply the rule that the sum of all the angles of the triangle is 180° on △ABC,

Substutitu (x-2) from m<A, (4x+1) for m<B and (4x+1) for m<x.

m<A+m<b+m<c=180
x-2+4x+1+4x+1=180
x+4x+4x-2+1+1=180
9x=180°
Divide both sides by 9

⇒ \(\frac{9 x}{9}=\frac{180}{9}\)

x=20

Substitute the value 20 for x in (x-2) to determine m<A

m<A = x-2

m<A=20-2

m<A=18

Substitute the value 20 for x in (4x+1) to determine m<B.
m<B =4x+1
m<B=4(20=+1
m<B=7=80+1
m<B=81°

As m∠C has the same value, m∠C = 81°.

The measures of the angles of △ABC are,

m∠A = 18°
m∠B = 81°
m∠C = 81°

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 2 Exercise 3 Answer

A statement of a student is given that the sum of two real numbers is always a rational number.

It is asked to determine if the student is correct and give a reason with an example.

To solve this question, it is important to know that every rational number is a real number. Real numbers also include irrational numbers. The sum of two rational or irrational numbers is always a rational number. Therefore, the given statement is correct.

For example, consider two real numbers \(\frac{1}{2} \text { and } \frac{3}{2}\)

Some of these two real numbers are, ​


⇒ \(\frac{1}{2}+\frac{3}{2}\) = \(\frac{4}{2}\)

⇒ \(\frac{1}{2}+\frac{3}{2}\) = 2

Here, 2 is a rational number.

The student is correct because the sum of two real numbers is always a rational number. The sum of two rational or irrational numbers is always a rational number. Therefore, the given statement is correct. For example, consider two real numbers \(\frac{1}{2} \text { and } \frac{3}{2}\)

The sum of these two real numbers is,

⇒ \(\frac{1}{2}+\frac{3}{2}\) = \(\frac{4}{2}\)

\(\frac{1}{2}+\frac{3}{2}\) = 2

Here, 2 is a rational number.

 

Page 3 Exercise 1 Answer

The statement is given that the sum of a number p and 12 is 34.

It is asked to identify the correct equation which represents this statement from the options given below.​

A. P − 12 = 34

B. 12P = 34

C. P + 12 = 34

D. P + 34 = 12

In the given statement the word ‘is’ represents equal to.

On the left side of it there is the phrase the sum of a number p and 12 which is represented as p + 12 and on the right side of it, 34 is given.

Therefore, put equal to sign between these two and obtain the required equation.

p + 12 = 34

Thus, option C is the correct answer.

The equation given in option A, P − 14 = 34 represents that the difference of p and 12 is 34.

The equation given in option B, 12P = 34 represents that the multiplication of p and 12 is 34.

The equation given in option D, P − 14 = 34 represents that the sum of p and 34 is 12.

The given statement is represented by the equation p + 12 = 34 and therefore, option C is the correct answer.

 

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 3 Exercise 3 Answer

It is given that the scale model of a skyscraper is 2ft tall and the scale of the model is 1in:7.2m.

It is asked to determine the height of the proposed skyscraper in meters.

To solve this question, first convert the model height from feet into inch as in the scale, inch unit is used. After that assume that the height of the skyscraper in meters is x and then according to the given scale, determine its value,

Covert the given height of the model from feet into inches.

1ft = 12in

2ft = 2 × 12

2ft = 24in

Assume the height of the proposed skyscraper as xm and solve for the variable using the given scale.

According to the scale 1in : 7.2m,1in of the model height represents 7.2m tall skyscraper.

The model height is 24in.

Therefore, the actual height of the proposed skyscraper is given by,

x = 24 × 7.2

x = 172.8m

The proposed skyscraper is 172.8m tall.

 

Page 4 Exercise 1 Answer

It is given that Tamika bought 6 apples and 1 juice.

It is required to find the cost of 1 apple if the cost of 1 juice is $1.25 and the total cost is $5.75

Let the cost of an apple be x. The sum of the cost of 6 apples and 1 juice is $5.75.

Therefore, the equation is formed as 6x + 1.25 = 5.75

Subtract 1.25 from both sides of the formed equation.​

Given: 6x+1.25=5.75

Subtract 1.25 from both sides

6x+1.25-1.25= 5.75-1.25

6x=4.50

divide both sides of an equation

⇒ \(\frac{6 x}{6}=\frac{4.50}{6}\)

x=0.75

Thus, the cost of 1 apple is $0.75.

Hence, option B is correct.

Let the cost of an apple be x. The sum of the cost of 6 apples and 1 juice is $5.75.

Therefore, the equation is formed as

Given: 6x+1.25=5.75

Subtract 1.25 from both sides

6x+1.25-1.25=5.75-1.25

6x=4.50

divide both sides of an equation

\(\frac{6 x}{6}=\frac{4.50}{6}\) ​

x=0.75

Thus, the cost of 1 apple is $0.75 which is not equal to $0.50.

Hence, option A is incorrect.

Option C

Let the cost of an apple be x. The sum of the cost of 6 apples and 1 juice is $5.75.

Therefore, the equation is formed as

Given: 6x+1.25=5.75

Subtract 1.25 from both sides

6x+1.25-1.25=5.75-1.25

6x=4.50

Divide both sides of an equation

\(\frac{6 x}{6}=\frac{4.50}{6}\) ​

x=0.75

Thus, the cost of 1 apple is $0.75 which is not equal to $1.00.\

Hence, option C is incorrect.

Option D

Let the cost of an apple be x. The sum of the cost of 6 apples and 1 juice is $5.75.

Therefore, the equation is formed as

Given: 6x+1.25=5.75

Subtract 1.25 from both sides

6x+1.25-1.25=5.75-1.25

6x=4.50

Divide both sides of an equation

​\(\frac{6 x}{6}=\frac{4.50}{6}\)

x=0.75

Thus, the cost of 1 apple is $0.75 which is not equal to $0.25.

Hence, option D is incorrect.

The correct answer is option B, the cost of 1 apple is $0.75.

Page 5 Exercise 1 Answer

An inequality is given as 2x 18.

It is required to solve inequality.

Divide both sides of inequality 2x 18 by 2. Since 2 is a negative number, therefore, the sign of inequality will get reversed.

⇒ \(\frac{-2 x}{-2} \leq \frac{-18}{-2}\)

x ≤ 9

Therefore, option
C is correct.

Option A

An inequality is given as 2x 18.

It is required to solve the inequality.

Divide both the sides of inequality 2x 18 by 2. Since 2 is a negative number, therefore, the sign of inequality will get reversed.

⇒ \(\frac{-2 x}{-2} \leq \frac{-18}{-2}\)

x9


Therefore, option
A is incorrect.

Option B

An inequality is given as 2x 18.

It is required to solve the inequality.

Divide both the sides of inequality 2x 182  by 2. Since 2 is a negative number, therefore, the sign of inequality will be reversed.

⇒ \(\frac{-2 x}{-2} \leq \frac{-18}{-2}\)

x9


Therefore, option
B sin-1 is incorrect.

Option D

An inequality is given as 2x 18.

It is required to solve the inequality.

Divide both the sides of inequality 2x 18 by 2. Since 2 is a negative number, therefore, the sign of inequality will get reversed.

⇒ \(\frac{-2 x}{-2} \leq \frac{-18}{-2}\)

x9

Therefore, option D is incorrect.

The option
C is correct, the given inequality is simplified as x 9.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 5 Exercise 2 Answer

An equation y = 3x + 6 and four ordered pairs (3,1), (−3,3), (0,6) and (6,0) are given.

It is required to find which ordered pair is the solution of the given equation.

To do so, substitute the given value of the ordered pair as x and y in the equation. Do this for all four ordered pairs and check which one maintains the equality.

For the order pair (0,6),

Substitute o for x in y=3x+6 and find.

y=3(0)+6

y=0+6

y=6

Hence, (0,6) is a solution. Therefore, option C is correct.

Substitute the given value of the ordered pair as x and y in the equation. Do this for all the other three ordered pairs.

Option A

For the order pair (0,6),

Substitute o for x in y=3x+6 and find y.

y=3(0)+6
y=0+6
y=6

Hence, (3,1) is not a solution. Therefore, option A is incorrect.

Option B

For the order pair(-3,3)
Substitute -3 for x in y=3x+6 and find y.

y=3(-3)+6
y=-9+6
y=-3

Hence, (−3,3) is not a solution. Therefore, option B is incorrect.

Option D

For the ordered pair (6,0)
Substitute 6 for x in y=3x+6 and find y.

​y=3(6)+6
y=18+6
y=24

Hence, (6,0)2 is not a solution. Therefore, option D is incorrect.

Option C is correct, the ordered pair which is a solution of the equation y = 3x + 6 is (0,6).

Page 5 Exercise 3 Answer

A function rule is given as h(x) = 2x + 6, where h is the height of the plant in centimeters after x weeks of growth.

It is required to graph the function and find the height of the plant after 9 weeks.

To do so, find the value of y if the value of x is assumed to be 1, 2, and 3.

Represent the ordered pairs obtained as a table and then plot the graph. From the graph, find the height of the plant after 9 weeks.

Given, h(x)=2x+6
letx=1
substitute 1 for x in h(x) = 2x+6 and find value of h

h(1)=2(1)+6
h=2+6
h=8

Hence, the ordered pair (1,8) is a solution of a given function.

Given, h(x)=2x+6
Letx=2
Substitute 2x for x in h(x) = 2x+6

h=2(2)+6
h=4+6
h=10

​Hence, the ordered pair (2,10) is a solution of a given function.

Given,h(x)=2x+6
Let x=3
substitute 3 for x in h(x)=2x+6

​h=2(3)+6
h=6+6
h=12

Hence, the ordered pair (3,12) is a solution of a given function.

Form a table for the obtained ordered pairs.

Standard Practices Page 5 Exercise 3 Answer Image 1

Plot the ordered pairs and join them through a line.

Standard Practices Page 5 Exercise 3 Answer Image 2

The function is plotted as,

Standard Practices Page 5 Exercise 3 Answer Image 2

Also, after 9 weeks the height of the plant is 24 cm.

Page 5 Exercise 4 Answer

Some ordered pairs are given as (0,40), (1,45), (2,50), (3,55), (4,60) and (5,65).

It is required to represent the ordered pairs as a table.

To do so, form a table with two rows one for x coordinates and the other for y coordinates for the given ordered pairs and then fill the table.

Form a table for the given ordered pairs.

Standard Practices Page 5 Exercise 4 Answer Image 1

The table for the given ordered pairs is,

Standard Practices Page 5 Exercise 4 Answer Image 1

Some ordered pairs are given as (0,40), (1,45), (2,50), (3,55), (4,60) and (5,65).

It is required to represent the ordered pairs as an equation.

To do so, assume a general equation of a line and substitute the value of given ordered pairs as x and y in the equation to find the relation between m and c.

Find the value of m and c, then substitute them in the general equation to find the required equation.

Let the equation be y = mx + c.

Then the ordered pair (0,40) will be the solution of the equation.

Let the equation be y=mx+c

for the ordered pair(0,40)

Substitute o for x and 40 for y in y=mx+x

40=m(0)+c
40=o+c
40=c

Then the ordered pair (1,45) will be the solution of the equation.

Substitute 1 for x, 45 for y, and 40 for c in y = mx + c.

Let the equation be y=mx+c
for the ordered pair (1,45)

Substitute 1 for x, 45 for y in y=mx+c

45=m(1)+40
45=m+40
45-40=m+40-40
5=m

Substitute 5 for m and 40 for c in y = mx + c.

y = 5x + 40

The given ordered pairs are represented in the equation as y = 5x + 40.

Some ordered pairs are given as (0,40), (1,45), (2,50), (3,55), (4,60) and (5,65).

It is required to represent the ordered pairs and the equation on a graph.

To do so, plot the equation on a graph and the given ordered pairs also.

Plot the equation y = 5x + 40 on a graph and the given ordered pairs also.

Standard Practices Page 5 Exercise 4 Answer Image 2

The given ordered pairs and the y = 5x + 40 is plotted as,

Standard Practices Page 5 Exercise 4 Answer Image 2

Given the ordered pairs (0,40), (1,45), (2,50), (3,55), (4,60) and (5,65).

It is required to describe a situation that the ordered pair might represent.

To do so, consider any two ordered pairs and represent them as an equation. Observing the equation, a situation can be determined that the ordered pairs represent.

Form an equation from the ordered pairs (0,40) and (1,45).

The ordered pairs (0,40) and (1,45)
The equation can be written as,

⇒ \(\begin{aligned}
& y=\frac{45-40}{1-0} \cdot x+c \\
& y=5 x+c
\end{aligned}\)

For (0,40)
C=40
The equation becomes y=5x+40

The equation can be written in the form v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t represents the time.

The ordered pairs may represent time and velocity at that time.

Page 6 Exercise 2 Answer

It is given that the slope of the line is 9 and passes through the point (−3,6).

It is required to find the equation of the line.

To do so, substitute the slope of the line and the point in the point-slope form equation. As a result, the equation of the line can be determined.

Substitute the slope of the line and the point in the point-slope form equation.

y − 6 = 9(x−(−3))

y − 6 = 9(x+3)

The equation of the line is y − 6 = 9(x+3).

So, the correct option is (A)y − 6 = 9(x+3).

It is given that the slope of the line is 9 and passes through the point (−3,6).

Option (B)

The given equation is y − 6 = 9(x−3).

Consider the point (−3,6) and substitute in the equation y − 6 = 9(x−3),

Given equation is y+3=9(x-6)
For the ordered pair (-3,6)

y+3=9(x-6)
6+3=9(-3-6)
9=9(-9)
9=-54

The values on both sides of the equation are different, so the option (B) is not correct.

Option (C)

The given equation is y + 3 = 9(x−6).

Consider the point (−3,6) and substitute in the equation y + 3 = 9(x−6),

Given equation is y+3=9(x-6)
for the ordered pair(-3,6)

y+3 =9(x-6)
6+3=9(-3-6)
9=9(-9)
9=-54

The values on both sides of the equation are different, so option (C) is not correct.

Option (D)

The given equation is y − 6 = −3(x−9).

Consider the point (−3,6) and substitute in the equation y − 6 = −3(x−9),

Given equation is y-6 =-3(x-9)
For the ordered pair (-3,6)
y-6=-3(x-9)
6-6=-3(-3-9)
0=-3(-12)
0=36

The values on both sides of the equation are different, so the option (D) is not correct.

The correct answer is Option (A)y − 6 = 9(x+3).

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 6 Exercise 3 Answer

It is given that sage earns 6$ per hour.

It is required to make a table and write an equation to show the relationship between the number of hours worked h and the wages earned w.

To do so, take the values of hours and multiply to that of wages earned per hour. If the amount earned per hour is a, then the amount earned for n hours can be calculated by the unitary method shown below,

Amount earned per hour = a

Amount earned for n hours = na.

Take the values of hours 1, 2, 3, 4 and multiply with that of wages earned per hour, 6$.

For 1 hour, the wage earned is 6⋅1$ = 6$.

For 2 hours, the wage earned is 6⋅2$ = 12$.

For 3 hours, the wage earned is 6⋅3$ = 18$.

For 4 hours, the wage earned is 6⋅4$ = 24$.

Make a table and write an equation to show the relationship between the number of hours worked h and the wages earned w.

Implement the above values in table. The table can be made as shown below,

Standard Practices Page 6 Exercise 3 Answer

The equation to show the relationship between the number of hours worked h and the wages earned w can be written as w = 6⋅h, where w represents the wages earned and h is the number of hours worked.

The table to show the relationship between the number of hours worked h and the wages earned w is shown below,

Standard Practices Page 6 Exercise 3 Answer

The equation to show the relationship between the number of hours worked h and the wages earned w is w = 6⋅h.

It is given that sage earns 6$ per hour.

It is required to find how many hours will sage need to work to earn 30$.

To do so, use the unitary method to convert the amount sage earns per hour into the amount sage needs to work to earn 30$. Find the number of hours sage needed to work to earn 1 $​​ and as a result, the number of hours sage need to work to earn 30$ can be determined.

Find the number of hours sage will need to work to earn 30$.

It is given that sage earns 6$ per hour,

To earn 6 ​​$​​, number of hour sage will need to work = 1 hour

To earn 1 $​​, number of hour sage will need to work = \(\frac{1}{6} \text { hour }\)

To earn 30 ​​$​​, number of hours sage will need to work = \(\frac{1}{6} \times 30 \text { hours }\)

Therefore to earn 30$, sage will need to work 5 hours.

Sage will need to work 5 hours to earn 30$.

Page 6 Exercise 4 Answer

Given the final purchase price of the item is $175.

It is required to find the possible price of the item before each discount.

To do so, add the discount and the purchase price of the item. As a result, the purchase price of the item. As a result, the purchase price of the item before each discount can be determined.

Find the price for a flat discount of $20 off on any purchase.

Price before discount = discount + final purchase price
price before discount = $20+$175
Price before discount = $195

Find the price for a flat discount of 20 off on total purchase
if the price before the discount is p,

⇒ \(\left(P-\frac{20}{100} \cdot P\right)=\$ 175\)

0.8p=$ 175

P=$ 218.75

The possible prices of the item before each discount is $195 and $218.75.

Given the final purchase price of the item is $175.

It is required to find the discount which represents a bigger saving in cost for the customer.

To do so, add the discount and the purchase price of the item. As a result, the purchase price of the item before each discount can be determined. Compare the two prices and the discount which represents a bigger saving in cost can be determined.

Find the price for a flat discount of $20 off on any purchase.

Price before discount = discount + final purchase price
price before discount = $20+$175
Price before discount = $195

Find the price for a flat discount of 20 off on total purchase
if the price before the discount is p,

\(\left(P-\frac{20}{100} \cdot P\right)=\$ 175\)

0.8p=$ 175

P=$ 218.75

A flat discount of 20% represents a bigger saving in cost for the customer.

Page 7 Exercise 1 Answer

Given four graphs as shown below,

(A)

Standard Practices Page 7 Exercise 1 Answer Image 1

(B)

Standard Practices Page 7 Exercise 1 Answer Image 2

(c)

Standard Practices Page 7 Exercise 1 Answer Image 3

(D)

Standard Practices Page 7 Exercise 1 Answer Image 4

It is required to determine which of the graph is not a function.

In the graph in option(C), the line is parallel to the y-axis.

A function is a relation in which each input has a single output.

In the graph in Option (C), for any value of y, x is the same.

For a single value of input x, the output y could be anything. Therefore the graph is not a function as the output y has more than one value.

The graph in Option(C) is not a function.

A function is a relation in which each input has a single output.

In the graph in option (A), (B) and (D), for every input, there is an output.

The graph in option (A), (B) and (D) are functions.

The correct answer is Option (C).

 

Page 7 Exercise 2 Answer

Given a graph shown below,

Standard Practices Page 7 Exercise 2 Answer Image 1

It is required to draw a line on the graph shown above that has the same slope as the line is drawn and passes through (−2,1).

To do so, find the slope of the line given in the graph. Use the slope and the given point to generate an equation and as a result, the required line can be drawn.

Find the equation of the given line.

From the graph x-intercept is 2 and the y-intercept is 4.

The equation can be written as

\(\frac{x}{2}+\frac{y}{4}=1\)

2x+y=1

From the equation, the slope of the line is -2

find the equation of the line for point (-2,1) and slope -2

The equation of the line can be written as,

y-1=(-2)(x-(-2))
y-1=-2x-4
y=-2x-4+1
y=-2x-3

The y-intercept is −3.

Use the point (−2,1) and y-intercept, -3 to draw the line as shown below,

Standard Practices Page 7 Exercise 2 Answer Image 2

The line that has the same slope as the line is drawn and passes through (−2,1) is shown below,

Standard Practices Page 7 Exercise 2 Answer Image 3

Given a graph shown below,

Standard Practices Page 7 Exercise 2 Answer Image 3

It is required to find the equation of the line that has the same slope as the line is drawn and passes through (−2,1).

To do so, find the slope of the line given in the graph. Use the slope and the given point to generate an equation and as a result, the equation of the line can be determined.

Find the equation of the given line.

From the graph, the x-intercept is 2 and the y-intercept is 4.

The equation can be written as

⇒ \(\frac{x}{2}+\frac{y}{4}=1\)

2x+y=1

From the equation, the slope of the line is -2

Find the equation of the line for point (-2,1) and slope -2

The equation of the line can be written as,

y-1=(-2)(x-(-2))
y-1=-2x-4
y=-2x-4+1
y=-2x-3

The equation of the line is y = −2x − 3.

Envision Algebra 1 Assessment Readiness Workbook Student Edition Chapter 1 Standards Practice 1 Page 7 Exercise 3 Answer

A graph and a set of equations are given in the question. The given equations are,

y – 2 = \(\frac{1}{2}(x-4)\)

y + 4 = 2(x-1)

y – 4 = \(\frac{1}{2}(x-2)\)

y – 1 = 2(x+4) and

y – 1 = \(\frac{1}{2}(x+4)\)

It is required to find which of the given equations matches the given graph.

To draw the graph use dynamic geometry software. Then

The given graph is,

Standard Practices Page 7 Exercise 3 Answer Image 1

The graph corresponding to the equation y – 2 = \(\frac{1}{2}(x-4)\) is given below.

Standard Practices Page 7 Exercise 3 Answer Image 2

The graph corresponding to the equation y + 4 = 2(x−1) is given below.

Standard Practices Page 7 Exercise 3 Answer Image 3

The graph corresponding to the equation y – 4 = \(\frac{1}{2}(x-2)\) is given below.

Standard Practices Page 7 Exercise 3 Answer Image 4

The graph corresponding to the equation y − 1 = 2(x+4) is given below.

Standard Practices Page 7 Exercise 3 Answer Image 5

The graph corresponding to the equation y – 1 = \(\frac{1}{2}(x+4)\) is given below.

Standard Practices Page 7 Exercise 3 Answer Image 6

From the above graphs it can be note that only the graphs in step 4 and 6 matches with the given graph.

Therefore the equations y – 4 = \(\frac{1}{2}(x-2)\), nd y – 1 = \(\frac{1}{2}(x+4)\) are the possible equations of the line drawn.

Standard Practices Page 7 Exercise 3 Answer Image 7

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2

Envision Math Grade 8 Volume 1 Chapter 8 Solve Problems Involving Surface Area And Volume

Page 423 Exercise 1 Answer

Given:

Solve Problems Involving Surface Area And Volume Page 423 Exercise 1 Answer
To find the similarity in the figures:

Apply the formula of the cylinder’s volume and the rectangular’s tank volume.
Solve Problems Involving Surface Area And Volume Page 423 Exercise 1 Answer Image

The similarity found in both shapes is the height of the shapes, which is the same.

They are different because both shapes have different volumes.

Given:

Solve Problems Involving Surface Area And Volume Page 423 Exercise 1 Answer

From the part(a), the volume of the rectangular tank is greater than the volume of the circular tank.

Since the volume of the rectangular tank is greater than the volume of the circular tank, the rectangular tank can hold more water.

Hence, Ricardo is correct.

Since the volume of the rectangular tank is greater than the volume of the circular tank, the rectangular tank can hold more water. Hence, Ricardo is correct.

Envision Math Grade 8 Surface Area And Volume Exercise 8.2 Answers

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 424 Essential Question Answer

Volume of cylinder is πr2h, where r = radius and h = height

Prisms and cylinders are similar because they both have two bases and a height.

The formula for the volume of a rectangular solid, V = Bh

V = Bh, can also be used to find the volume of a cylinder when area of base is given.

When area of base is given volume for both cylinder and prism is V = Bh

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2

Page 424 Try It Answer

Given: area = 78.5in2 and height = 11 in

To find: volume of cylinder

We will use the product of area and height to find the volume.

As we know V = Bh then putting all the values

V = 78.5 × 11

= 863.5 in3

Therefore, volume is given as 863.5in3

 

Page 425 Try It Answer

Given: a cylindrical planter with a base diameter of 15 inches and 5,000 cubic inches is the volume

To find: height of the planter

We will use the formula of volume of cylinder and find the value.

Volume of cylinder is πr2h, where r = radius and h = height
Solve Problems Involving Surface Area And Volume Page 425 Try It Answer

Height of planter is approximately 28 in.

Envision Math Grade 8 Volume 1 Student Edition Chapter 8 Exercise 8.2

Page 426 Exercise 1 Answer

Volume of cylinder is πr2h, where r = radius and h = height

Prisms and cylinders are similar because they both have two bases and a height.

The formula for the volume of a rectangular solid V = Bh

V = Bh can also be used to find the volume of a cylinder when area of base is given.

When area of base is given volume for both cylinder and prism is V = Bh

 

Page 426 Exercise 2 Answer

As we know the volume is the product of cross sectional area and height.

Cylinder has a circular base so its area is given as πr2

Now for height h, volume is πr2 × h

Two measurements you need to know to find the volume of a cylinder is radius and height.

 

Page 426 Exercise 3 Answer

Given: Cylinder A has a greater radius than Cylinder B.

To find: Cylinder A necessarily have a greater volume than Cylinder B?

Volume of cylinder is πr2h, where r = radius and h = height

Volume depends on radius and height both so it is not necessary.

Cylinder A does not necessarily have a greater volume than Cylinder B.

Solutions For Envision Math Grade 8 Exercise 8.2

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 426 Exercise 4 Answer

Given: area is 4πmm2 and height is 10 mm

To find: Volume of Cylinder

We will use the formula and put the values to find volume

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 426 Exercise 4 Answer

Volume of the cylinder is 40π mm3

 

Page 426 Exercise 5 Answer

Given: radius is 10 ft and volume is 314 ft3

To find: height of Cylinder

We will use the formula and put the values to find height.

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 426 Exercise 5 Answer

Volume of cylinder is one feet.

 

Page 426 Exercise 6 Answer

Given: radius is 4 cm and circumference is 22.4 cm

To find: Volume of Cylinder

First we will find the radius from circumference.

We will use the formula and put the values to find volume.
Solve Problems Involving Surface Area And Volume Page 426 Exercise 6 Answer

Volume of the cylinder is 162.8 cm3

 

Page 427 Exercise 7 Answer

Given: radius is 5 cm and height is 2.5 cm

To find: Volume of Cylinder

We will use the formula and put the values to find volume

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 427 Exercise 7 Answer

Volume of cylinder is 196.25 cm3

 

Page 427 Exercise 9 Answer

Given: height is 1 in and volume is 225π in3

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 427 Exercise 9 Answer

Radius is calculated as 15 in.

Envision Math Exercise 8.2 Step-By-Step Solutions Grade 8

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 427 Exercise 10 Answer

Given: height is 8.1 cm and volume is 103 cm3

To find: Radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 427 Exercise 10 Answer

Radius of the bottle is 2.01 cm.

 

Page 427 Exercise 11 Answer

Given: height is 3 in and radius is 4 in

To find: Volume of Cylinder

We will use the formula and put the values to find volume.

As we know that volume is the product of base area and height.

Putting the values in formula

V = π × 42 × 3

= 48π in3

Volume of the cylinder is 48π in3

Given: height is 3 in and radius is 4 in

To find: is the volume of a cylinder, which has the same radius but twice the height, greater or less than the original cylinder?

We will use the formula and put the values to find radius.

First we will write the original volume and then find the new one.

As we know that volume is the product of base area and height.

The original volume is 48π in3

Now h′ = 2h which will be 6 in

Putting the values in formula

V′ = π × 42 × 6

= 96π in3

Clearly 98π in3 > 48π in3 therefore, volume of second cylinder is greater.

 

Page 428 Exercise 13 Answer

Given: height is 11.7 in and volume is 885 in3

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Solve Problems Involving Surface Area And Volume Page 428 Exercise 13 Answer

Radius of the cylinder is 4.91 in

Given: height is 11.7 in and volume is 885 cubic inches

To find: If the height of the cylinder is changed, but the volume stays the same, then how will the radius change

We will use the formula of volume and find the dependency of volume.

As we know that volume is the product of base area and height.

That means it is dependent on both radius and height.

V α r2h

If the height of the cylinder is changed, but the volume stays the same, then radius will decrease.

Solve Problems Involving Surface Area And Volume Envision Math Solutions Exercise 8.2

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 428 Exercise 14 Answer

Given: height is 20.7 cm and diameter is 6.9 cm

To find: Volume of Cylinder

First we will find the radius.

We will use the formula and put the values to find volume.
Solve Problems Involving Surface Area And Volume Page 428 Exercise 14 Answer

Volume of the cylinder is 773.6 cm3

 

Page 428 Exercise 15 Answer

Given: height is 21 in, inner radius is 3 in and outer radius is 5 in

To find: Volume of material

We will use the formula and put the values to find radius.

We will find volume with both the radii and the difference will be required volume

As we know that volume is the product of base area and height.

Putting the values in formula to find outer volume

VR = π × 5 × 5 × 21

VR = 1648.50

As we know that volume is the product of base area and height.

Putting the values in formula for finding volume of inner volume

Vr = π × 3 × 3 × 21

Vr = 593.46

Difference in volumes is

VR − Vr = 1648.50 − 593.46

= 1055.04 in3

Volume of the required material is 1055.04in3

Envision Math Grade 8 Chapter 8 Worksheet Solutions Exercise 8.2

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 428 Exercise 16 Answer

Given: height is 21 cm and volume is 1029πcm3

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Solve Problems Involving Surface Area And Volume Page 428 Exercise 16 Answer

Radius of the Cylinder is 3.9 cm

 

Page 428 Exercise 17 Answer

Given: height is 12 yd and diameter is 7 yd

To find: Volume of Cylinder

First we will find the radius.

We will use the formula and put the values to find radius.

Radius is given as \(\frac{\mathrm{d}}{2}=\frac{7}{2}\)

As we know that volume is the product of base area and height.
Solve Problems Involving Surface Area And Volume Page 428 Exercise 17 Answer

Volume of cylinder is 147π cubic yards

Envision Math Grade 8 Chapter 8 Exercise 8.2 Explained

Page 429 Exercise 1 Answer

A three-dimensional object is in three dimensions having length, width and height.

Example: Cubes, prisms, pyramids, spheres, cones, and cylinders are all examples of three-dimensional objects.

Volume is the product of length, width and height.

Surface areas is the sum of products of length, width and height taken two at a time.

Surface area is a two-dimensional measure.

Volume is a three-dimensional measure.

 

Page 429 Exercise 3 Answer

Given: A figure of cylinder containing a cone

To find: how much cardboard was used to make the package

First we will find the radius.

Then with the help of tip we will find the surface area.

Radius of the cone is \(\frac{d}{2}\) = 10cm

Putting all the values in the formula

2πrh + 2πr2

S = 2πrh + 2πr2

S = 2π.10.33 + 2π.33.33

= 860π cm2

Surface area for cylinder is 860π cm2

Free Solutions For Envision Math Grade 8 Exercise 8.2

Envision Math Grade 8 Volume 1 Student Edition Solutions Chapter 8 Solve Problems Involving Surface Area And Volume Exercise 8.2 Page 429 Exercise 5 Answer

Given: A figure of sphere with radius as 3 ft

To find: Surface area

Then with the help of tip we will find the surface area.

Putting all the values in the formula of surface area

Solve Problems Involving Surface Area And Volume Page 429 Exercise 5 Answer

Surface area of the given sphere is 36π ft2

How To Solve Envision Math Grade 8 Surface Area And Volume Problems Exercise 8.2 

Page 429 Exercise 6 Answer

Given: Volume is 400πcm3 and diameter is 10 cm

To find: Which option has the correct height

We will use the formula and put the values to find height.

Solve Problems Involving Surface Area And Volume Page 429 Exercise 6 Answer Image 1

Solve Problems Involving Surface Area And Volume Page 429 Exercise 6 Answer Image 2
So the height does not match with any of the option (A), (C) or (D).

Correct option is (B).