Carnegie Learning Algebra I Student Text Volume 1 3rd Edition Chapter 1 Quantities And Relationships
Page 4 Problem 1 Answer
Given: You may have purchased ice, gone grocery shopping, selected music, made food, or even cleaned in preparation.
To describe how you can determine which quantity is the independent quantity and which quantity is the dependent quantity in any problem situation.Using the method of the graph.
The dependent quantity is the quantity that charges when the other quantity changes and the independent quantity is the quantity that causes the dependent quantity to charge.
The dependent quantity is the quantity that charges when the other quantity changes and the independent quantity is the quantity that causes the dependent quantity to charge.
Page 5 Problem 2 Answer
Given: Scenario.To determine the independent and dependent quantities.Using the method of the graph.
Something’s Fishy:
In this scenario, the two quantities are time and gallons in the tank since the given rate is that the water drains from the tanks at a constant rate of 10 gallons per minute. The number of gallons left in the tank depends on how many minutes the tank has been draining.
Therefore:
Independent quantity: time in minutes.
Dependent quantity: number of gallons remaining in the tank.
Smart Phone, but Is It a Smart Deal?
In this scenario, the two quantities are the number of weeks and the amount of interest you owe to your cousin.
As each week goes by, the amount of interest doubles so the amount of interest depends on the number of weeks.
Therefore:
Independent quantity: number of weeks.
Dependent quantity: the amount of interest in dollars.
Can’t Wait to Hit the Slopesl:
In this scenario, it is given that the ski lift is ascending up the mountain at a steady rate of 83 feet per minute.
The two quantities are then the height of the ski lift and the number of minutes that have passed.
The height of the ski lift depends on the number of minutes that have passed. Therefore:
Independent quantity: number of minutes.
Dependent quantity: height of the ski lift in feet.
It’s Magic:
In this scenario, the two quantities are the length of the remaining piece of rope and the total number of cuts.
The more times he cuts the rope, the short the remaining piece of rope gets so the length of the remaining piece of the rope depends on the total number of cuts.
Therefore:
Independent quantity: total number of cuts.
Dependent quantity: length of the remaining piece of rope in feet.
Baton Twirling:
In this scenario, the two quantities are the height of the baton and the amount of time since she needs the baton to be in the air for 2
seconds before she catches it when it comes back down. The height of the baton depends on the number of seconds that have passed.
Therefore:
Independent quantity: number of seconds.
Dependent quantity: height of the baton in feet.
Music Club:
In this scenario, it is given that Jermaine pays $1 per song.
This rate means the two quantities are the amount of money Jermaine has paid for songs and the number of songs he has purchases.
The amount of money he pays depends on the number of songs he purchases.
Therefore:
Independent quantity: number of songs.
Dependent quantity: total cost in dollars.
A Trip to School:
In this scenario, the two quantities are time in minutes and the distance she has traveled since her distance from home is changing.
The distance she is from home increases as the number of minutes increases so her distance depends on the number of minutes.
Therefore:
Independent quantity: time in minutes.
Dependent quantity: distance in miles.
Jelly Bean Challenge:
In this scenario, the two quantities are the number of jelly beans that each person guesses and how many jelly beans each person’s guess was off by since these are the two quantities that Mr. Wright is keeping track of.
A person’s guess determines how far off their guess is from the actual number of jelly beans so how far off their guess depends on how many jelly beans they guess.
Therefore:
Independent quantity: number of jelly beans a person guesses.
Dependent quantity: how many jelly beans the guess is off by.
Something’s Fishy: independent: time in minutes, dependent quantity: number of gallons remaining in the tank.
Smart Phone, but Is It a Smart Deal?: independent: number of weeks, dependent: amount of interest in dollars.
Can’t Wait to Hit the Slopesl: independent: number of minutes, dependent: height of the ski lift in feet.
It’s Magic: independent: total number of cuts, dependent: length of remaining piece of rope in feet.
Baton Twirling: independent: number of seconds, dependent: height of the baton in feet.
Music Club: independent: number of songs, dependent: total cost in dollars.
A Trip to School: independent: time in minutes, dependent: distance in miles.
Jelly Bean Challenge: independent: number of jelly beans a person guesses, dependent: how many jelly beans the guess is off by.
Page 15 Problem 3 Answer
Given: A graph with the appropriate problem.
To find the similarities do you notice in the graphs? Using the method of the graph.
Graphs A and H have constant rates of change. Graphs A and B are only increasing. Graphs D and H are only decreasing.
Graphs C and F both have an increasing portion of the graph and a decreasing portion of the graph.
Graphs E and G are both increasing, constant, and then increasing again.
Graphs A and H have constant rates of change. Graphs A and B are only increasing.
Graphs D and H are only decreasing.
Graph C and F both have an increasing portion of the graph and a decreasing portion of the graph.
Graphs E and G are both increasing, constant, and then increasing again.
Page 15 Problem 4 Answer
Given: A graph with the appropriate problem situation.
To find what differences do you notice in the graphs Using the method of the graph.
Some graphs have constant rates of changes while others have variable rates of change.
Some graphs are either only increasing or decreasing while others have both increasing and decreasing portions.
Some graphs have constant rates of change while others have variable rates of change.
Page 15 Problem 5 Answer
Given: Independent and dependent quantities.
To find how did you label the independent and dependent quantities in each graph?Using the method of the graph.
For each graph, the independent quantity was labeled on the x axis and the dependent quantity was labeled on the y−axis.
For each graph, the independent quantity was labeled on the x axis and the dependent quantity was labeled on the y axis.
Page 15 Problem 6 Answer
Given: Analyze each graph from left to right.To describe any graphical characteristics you notice.Using the method of the graph.
Graph A is only increasing and has a constant rate of change. It has an x – and y-intercept at (0,0). It has no negative y-values.
Graph B is only increasing and has a variable rate of change. It has a y-intercept at(0,1) and no x-intercept. It has only positive y-values.
Graph C is decreasing with a constant rate of change and then increasing with a constant rate of change.
It has both an x-intercept and y-intercept. It doesn’t have any negative y-values.
Graph D is only decreasing with a variable rate of change. It has a y -intercept at (0,20) and no x-intercept.
It has only positive y-values.
Graph E is increasing with a constant rate of change, then increasing at a greater constant rate of change, then constant with a rate of change of 0, and then increasing again at a constant rate of change.
It has an x – and y-intercept at (0,0). It doesn’t have any negative y-values.
Graph F is increasing at a variable rate of change and then decreasing at a variable rate of change.
It has a y-intercept at (0,6), no x-intercept, and a maximum at (1,22).
It only has y-values that are greater than or equal to 6.
Graph G is increasing at a constant rate of change, then constant with a rate of change of 0, and then increasing again at a constant rate of change.
It has an x – and y-intercept at (0,0). It doesn’t have any negative y−values.
Graph H is decreasing at a constant rate of change. It has a y intercept at (0,20).
If the graph continued for larger values of x, it would have an x-intercept at (20,0).
It doesn’t have any negative values of y.
Graph A: increasing with constant rate of change, x – and y intercept of (0,0), no negative y-values
Graph B: increasing with variable rate of change, no x-intercept, y intercept at (0,1), only positive y-values
Graph C: decreasing and then increasing with constant rates of changes, has an x-intercept and y−intercept, no negative y-values
Graph D: decreasing with variable rate of change, no x-intercept, y -intercept at(0,20), only positive y-values.
Graph E: increasing, constant, and then increasing with constant rates of changes, x – and y−intercept at (0,0), no negative y−values
Graph F: increasing and then decreasing with variable rates of changes, no x-intercept, y−intercept at (0,6), maximum at (1,22), only y-values greater than or equal to 6.
Graph G: increasing, constant, and then increasing with constant rates of changes, x- and y−intercept at(0,0). no negative y-values
Graph H: decreasing with constant rate of change, x-intercept at (20,0),y-intercept at (0,20), no negative y-values.
Page 16 Problem 7 Answer
Given: Smart Phone, but Is It a Smart Deal? and Music Club.
To compare the graphs for each scenario given and describe any similarities and differences you notice.Using the method of the graph.
Smart Phone, but Is It a Smart Deal? (graph B) and Music Club (graph A) are both only increasing. Graph A is increasing at a constant rate.
Smartphone, but Is It a Smart Deal (graph B) and Music Club (graph A) are both only increasing. Graph A is increasing at a constant rate though while Graph B is increasing at a variable rate.
Page 16 Problem 8 Answer
Given: Something’s Fishy and It’s MagicTo compare the graphs for each scenario given and describe any similarities and differences you notice.
Using the method of the graph.
Something’s Fishy (graph H) and It’s Magic (graph D) are both only decreasing.
Graph D is decreasing at a variable rate while Graph H is decreasing at a constant rate.
Something’s Fishy (graph H) and It’s Magic (graph D) are both only decreasing.
Graph D is decreasing at a variable rate while Graph H is decreasing at a constant rate.
Page 16 Problem 9 Answer
Given: Baton Twirling and Jelly Bean ChallengeTo compare the graphs for each scenario given and describe any similarities and differences you notice.
Using the method of the graph.
Baton Twirling (graph C) and Jelly Bean Challenge (graph F) both have sections of increasing and decreasing.
Graph C decreasing and then increasing while Graph F increases and then decreases.
Graph C has constant rates of change while Graph F has variable rates of change.