# 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 39  and the height of the cylinder is 17 . If the volume of the solid is 150 pi, what is the area of the base of the cylinder?

Apr 7, 2017

The area of the base of the cylinder is $5 \pi$.

#### Explanation:

The formula for volume of a cone is:
$V = \pi {r}^{2} \frac{h}{3}$

The formula for volume of a cylinder is:
$V = \pi {r}^{2} h$

Therefore the formula for the total volume (${V}_{T}$) of the given solid is:

${V}_{T} = \pi {r}^{2} {h}_{1} / 3 + \pi {r}^{2} {h}_{2}$ (where ${h}_{1} =$height of cone, and ${h}_{2} =$height of cylinder. The radius, $r$, is the same for both.)

${V}_{T} = \pi {r}^{2} \left({h}_{1} / 3 + {h}_{2}\right)$

We need to calculate $r$ in order to calculate the area of the base of the cylinder, hence we fill in the data given.

$150 \pi = \pi {r}^{2} \left(\frac{39}{3} + 17\right)$

We cancel the like term ($\pi$) on each side.

$150 \cancel{\pi} = \cancel{\pi} {r}^{2} \left(\frac{39}{3} + 17\right)$

$150 = {r}^{2} \left(13 + 17\right)$

$150 = {r}^{2} \times 30$

Divide both sides by $30$.

$\frac{150}{30} = {r}^{2}$

$5 = {r}^{2}$

The formula of area of the base of a cylinder (a circle) is:

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

$A = \pi \times 5$

$A = 5 \pi$