# 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 \(k\).

- \(k=\frac{1}{4}\)
- \(k=0.4\)
- \(k=1\)
- \(k=1.2\)
- \(k=\frac{5}{3}\)

### 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.”

- What might Kiran have done wrong?
- What is the volume of the image?

### Problem 4

A parallelogram has an area of 10 square feet.

- 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 - Plot the points in the table on coordinate axes and connect them to create a smooth curve.

### 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.

\((0,0)\)

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

\((1,1)\)

\((4,1)\)

\((8,2)\)

### 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?

### 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.

### Problem 8

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

### Problem 9

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