Unit 1, Lesson 8
The Root of the Problem
Integrated Math 3
Student Lesson Summary
Suppose a prism has a volume of 5 cubic units. The prism is dilated, and the resulting solid has volume 320 cubic units. If we want to find the scale factor that was used in the dilation, we start by dividing the new volume, 320 cubic units, by the original volume, 5 cubic units, to find that the prism’s volume increased by a factor of 64. We know that when a solid is dilated by a scale factor , the volume is multiplied by . So, we need to find the number whose cube is 64. This number is called the cube root of 64 and is written . We know because . That is, the prism was dilated by a scale factor of .
We can create a graph that shows the relationship between the volume, , of a dilated solid and the scale factor, , needed to achieve it. Let’s use the prism with volume 5 cubic units as an example. Create a table of values, and then plot the points and connect them with a smooth curve.
| dilated volume in cubic units () | scale factor () |
|---|---|
| 0 | 0 |
| 5 | 1 |
| 40 | 2 |
| 135 | 3 |
| 320 | 4 |
This graph represents the equation . The graph rises relatively steeply from but quickly flattens out.
We can also find the scale factor of dilation if we know the surface areas of the original and dilated solids. Suppose a cylinder has surface area 35 square units, and is dilated resulting in a surface area of 218.75 square units. Divide the numbers to find that the surface area increased by a factor of 6.25. Take the square root of 6.25 to conclude that the solid was dilated by a factor of 2.5.