## 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

**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.

**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.

**Page 32 Exercise 1 Answer**

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

The area of the square is A=100m^{2}

The side of the square will be 10m.

**Page 32 Exercise 2 Answer**

We need to solve x^{3} = 64

The value of x = 4

**Page 33 Exercise 3 Answer**

We need to solve a^{3} = 11

The solutions are c=√27 and c=−√27

**Page 32 Exercise 1 Answer**

We need to explain why are there two possible solutions to the equation s^{2 }= 100 And also, we have to explain why only one of the solutions is valid in this situation.

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

**Page 34 Exercise 2 Answer**

Suri solved the equation x^{2} = 49 and found that x=7

We need to find the error Suri made.

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 x^{2} = 17 is irrational.

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 = s^{3}

The length of each edge will be 3cm.

**Page 34 Exercise 7 Answer**

We need to solve x^{3} = −215

The solution is x = \(-\sqrt[3]{215}\)

**Page 35 Exercise 9 Answer**

We need to solve a^{3} = 216

The solution is a = 6

**Page 35 Exercise 12 Answer**

We need to solve y^{2} = 81

The solutions are y = +9 and y = −9

**Page 35 Exercise 13 Answer**

We need to solve w^{3} = 1000

The solution is w = 10

**Page 35 Exercise 14 Answer**

The area of a square garden is given as A=121ft^{2}. We need to find how long is each side of the garden.

The length of the side of the garden is 111ft

**Page 35 Exercise 15 Answer**

We need to solve b^{2} = 77

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

**Page 36 Exercise 19 Answer**

Given that, Manolo says that the solution of the equation g^{2} = 36 is g = 6 because 6 × 6 = 36

Check whether Manolo’s reasoning is complete or not

Therefore, Manolo’s reasoning is wrong.

**Page 36 Exercise 20 Answer**

We need to solve \(\sqrt[3]{-512}\)

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.

**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.

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 6859ft^{3
}

The hardwood floor costs 3610 dollars.

**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.

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.

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