# Lesson 6

Scaling Solids

• Let’s see how the surface area and volume of solids change when we dilate them.

### Problem 1

It takes 2 ounces of paint to completely cover all 6 sides of a rectangular prism box which holds 15 cups of sugar. Double the dimensions of the box. Approximately how much paint would the new box need? How much sugar would it hold?

### Problem 2

A solid with volume 12 cubic units is dilated by a scale factor of $$k$$. Find the volume of the image for each given value of $$k$$.

1. $$k=\frac{1}{4}$$
2. $$k=0.4$$
3. $$k=1$$
4. $$k=1.2$$
5. $$k=\frac{5}{3}$$

### Problem 3

A solid’s volume is 10 cubic inches. The solid is dilated by a scale factor of 3.5. Kiran says, “I calculated the volume of the image as 35 cubic inches, but I don’t think that’s right.”

1. What might Kiran have done wrong?
2. What is the volume of the image?

### Problem 4

A parallelogram has an area of 10 square feet.

1. Complete the table that shows the relationship between the dilated area ($$x$$) and the scale factor ($$y$$).
 dilated area in square feet scale factor 0 40 160 360 640
2. Plot the points in the table on coordinate axes and connect them to create a smooth curve.
(From Unit 5, Lesson 5.)

### Problem 5

A figure has an area of 4 square units. The equation $$y=\sqrt{\frac{x}{4}}$$ represents the scale factor of $$y$$ by which the solid must be dilated to obtain an image with area $$x$$ square units. Select all points which are on the graph representing this equation.

A:

$$(0,0)$$

B:

$$\left(1,\frac12\right)$$

C:

$$(1,1)$$

D:

$$(4,1)$$

E:

$$(8,2)$$

(From Unit 5, Lesson 5.)

### Problem 6

Tyler is designing a banner that will welcome people to a festival. The design for the banner has an area of 1.5 square feet. The actual banner will be a dilation of the design by a factor of 5. What will the area of the actual banner be?

(From Unit 5, Lesson 4.)

### Problem 7

The horizontal cross sections of this figure are dilations of the bottom rectangle using a point above the rectangle as a center and scale factors from $$\frac12$$ to 1. Sketch an example of a cross section that is created from using a scale factor of $$\frac34$$. Label the dimensions of the cross section that you sketch.

(From Unit 5, Lesson 3.)

### Problem 8

Technology required. A regular hexagon is inscribed in a circle of radius 1 inch. What is the area of the shaded region?

(From Unit 4, Lesson 10.)

### Problem 9

Two distinct lines, $$\ell$$ and $$m$$, are each perpendicular to the same line $$n$$.  Explain why $$\ell$$ and $$m$$ are parallel lines.

(From Unit 1, Lesson 6.)