We have been given two similar solids.

We want to find the volume of the larger solid. Let's begin by finding the scale factor from the larger solid to the smaller solid. Notice that the diameter of the base of the larger solid is

16 cm,

so its radius is

8 cm .

Therefore, the ratio of the radii of their bases will give us the scale factor.

Scale Factor \frac{6}{8} = \frac{3}{4} \Leftrightarrow 3 : 4

The scale factor of the solids is

3 : 4 .

Notice that we have been given just enough information to find the volume of the smaller solid. Therefore, let's first find the volume of the smaller solid. As we can see, the solids consist of a cylinder and a cone.

We will find the volume of the cylinder and the volume of the cone separately.

Cylinder : Rectangles : π (6)^{2} (7.2) \approx 814.3 cm^{3} \frac{π ( 6 ) ^{2} ( 5 . 7 )}{3} \approx 214.9 cm^{3}

Then we can add them together to find the volume of the smaller solid.

814.3 + 214.9 \approx 1029.2 cm^{3}

Next, we will recall Theorem

12.1

to relate the scale factor to the ratio of the volumes.

Theorem $12.1$
If two similar solids have a scale factor of $a : b,$ then the ratio of the surface areas is $a^{2} : b^{2},$ and the ratio of the volumes is $a^{3} : b^{3} .$

Exercise

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