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Volume Similar Solids

# Volume Similar Solids

The figures below are similar.

What is the volume of the smaller cone?

The following proportion applies to similar solids:

$(\frac{a}{b})^3=\frac{V_1}{V_2}$ where $\frac{a}{b}$ is the ratio of the corresponding dimensions, and $\frac{V_1}{V_2}$ is the ratio of the volumes.

Find the cube of the ratio of the corresponding dimensions:

$(\frac{a}{b})^3=(\frac{2}{5})^3=\frac{8}{125}$

Find the ratio of the volumes:

$\frac{V_1}{V_2}=\frac{V_1}{8,000}$

Use these two ratios to set up a proportion and solve for V1.

 $\frac{8}{125}$
=
 $\frac{V_1}{8,000}$

 8 × 8,000
=
125
 V1
Find the cross products

64,000 =
125
 V1
Simplify

 64,000 ÷ 125
=
125
 V1
÷ 125
Divide both sides by 125

512 =
 V1

The volume of the smaller cone is 512 cubic metres.

The figures below are similar.
What is the volume of the smaller rectangular pyramid?

 2 yd 6 yd 540 yd