# Lesson 7

The Root of the Problem

- Let’s look at relationships between volumes, areas, and scale factors using graphs and situations.

### Problem 1

A solid with volume 8 cubic units is dilated by a scale factor of \(k\) to obtain a solid with volume \(V\) cubic units. Find the value of \(k\) which results in an image with each given volume.

- 216 cubic units
- 1 cubic unit
- 1,000 cubic units

### Problem 2

A solid has volume 7 cubic units. The equation \(k=\sqrt[3]{\frac{V}{7}}\) represents the scale factor of \(k\) by which the solid must be dilated to obtain an image with volume \(V\) cubic units. Select **all** points which are on the graph representing this equation.

\((0,0)\)

\((1,1)\)

\((1,7)\)

\((7,1)\)

\((14,2)\)

\((49,2)\)

\((56,2)\)

\((27,3)\)

### Problem 3

A solid with surface area 8 square units is dilated by a scale factor of \(k\) to obtain a solid with surface area \(A\) square units. Find the value of \(k\) which leads to an image with each given surface area.

- 512 square units
- \(\frac{1}{2}\) square unit
- 8 square units

### Problem 4

It takes \(\frac18\) of a roll of wrapping paper to completely cover all 6 sides of a small box that is shaped like a rectangular prism. The box has a volume of 10 cubic inches. Suppose the dimensions of the box are tripled.

- How many rolls of wrapping paper will it take to cover all 6 sides of the new box?
- What is the volume of the new box?

### Problem 5

A solid with volume 8 cubic units is dilated by a scale factor of \(k\). Find the volume of the image for each given value of \(k\).

- \(k=\frac{1}{2}\)
- \(k=0.6\)
- \(k=1\)
- \(k=1.5\)

### Problem 6

A figure has an area of 9 square units. The equation \(y=\sqrt{\frac{x}{9}}\) represents the scale factor of \(y\) by which the solid must be dilated to obtain an image with area of \(x\) square units. Select **all** points which are on the graph representing this equation.

\((0,0)\)

\((1,1)\)

\((1,3)\)

\((3,1)\)

\((9,1)\)

\((9,3)\)

\((18,2)\)

\((36,2)\)

### Problem 7

Noah edits the school newspaper. He is planning to print a photograph of a flyer for the upcoming school play. The original flyer has an area of 576 square inches. The picture Noah prints will be a dilation of the flyer using a scale factor of \(\frac14\). What will be the area of the picture of the flyer in the newspaper?

### Problem 8

Angle \(S\) is 90 degrees and angle \(T\) is 45 degrees. Side \(ST\) is 3 feet. How long is side \(SU\)?