# A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is 9  and the height of the cylinder is 12 . If the volume of the solid is 15 pi, what is the area of the base of the cylinder?

Aug 16, 2017

$\pi$

#### Explanation:

The base of a cylinder is a circle. The area of a circle is $\pi {r}^{2}$; however, we do not know the radius. To find $r$, we can use the information given in the problem.

The volume of a cone is $\frac{1}{3} \pi {r}^{2} h$, where $r$ is the radius and $h$ is the height. We know the height is $9$, so we can say

${V}_{\text{cone}} = \frac{1}{3} \pi {r}^{2} \cdot 9 = 3 \pi {r}^{2}$

The volume of a cylinder is $\pi {r}^{2} h$, and we know the height is $12$.

${V}_{\text{cylinder}} = \pi {r}^{2} \cdot 12 = 12 \pi {r}^{2}$

We also know that the volume of the cone plus that of the cylinder is equal to $15 \pi$.

${V}_{\text{cone" + V_"cylinder}} = 15 \pi$

$3 \pi {r}^{2} + 12 \pi {r}^{2} = 15 \pi$

$15 \pi {r}^{2} = 15 \pi$

Dividing by $15 \pi$ on both sides, we get

${r}^{2} = 1$

$r = 1$

So, the area of the base of the cylinder is

$\pi {r}^{2} = \pi \cdot {1}^{2} = \pi$