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
6/8=3/4 ⇔ 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:& π(6)^2 (7.2)≈ 814.3 cm^3
Rectangles:& π(6)^2(5.7)/3≈ 214.9 cm^3
Then we can add them together to find the volume of the smaller solid.
814.3+214.9≈ 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.
By this theorem we can find the area scale factor of the solids.
c|c
Scale Factor & Ratio of the Volumes
3:4 &27:64
From here, we can write a proportion to find the volume of the larger solid, V.
1029.2/V=27/64
Finally, let's solve this proportion for V.
