# 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 12  and the height of the cylinder is 18 . If the volume of the solid is 72 pi, what is the area of the base of the cylinder?

Sep 7, 2016

The area of the base is $\frac{36}{11} \pi$.

#### Explanation:

Start by drawing a diagram.

The formula for volume of a cone is $V = \frac{1}{3} {r}^{2} h \pi$ and the formula for volume of a cylinder is $V = \pi {r}^{2} h$.

Let ${V}_{t}$ denote the total volume.

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

$72 \pi = \pi {r}^{2} h + \frac{1}{3} {r}^{2} h \pi$

$72 \pi = 18 \pi {r}^{2} + 12 \left(\frac{1}{3} {r}^{2} \pi\right)$

$72 \pi = 18 \pi {r}^{2} + 4 {r}^{2} \pi$

$72 \pi = 22 \pi {r}^{2}$

$\frac{36}{11} = {r}^{2}$

$r = \sqrt{\frac{36}{11}}$

$r = \frac{6}{\sqrt{11}}$

The formula for area of a circle is $A = \pi {r}^{2}$, so the area of the base is $A = {\left(\frac{6}{\sqrt{11}}\right)}^{2} \pi = \frac{36}{11} \pi$.

Hopefully this helps!