# 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 33  and the height of the cylinder is 14 . If the volume of the solid is 225 pi, what is the area of the base of the cylinder?

Jul 4, 2016

$\frac{75 \pi}{13}$

#### Explanation:

Assume the radius of the cylinder/cone as r, height of cone as ${h}_{1}$, height of cylinder as ${h}_{2}$

Volume of the cone part of solid = $\frac{\pi \cdot {r}^{2} \cdot {h}_{1}}{3}$

Volume of the cylinder part of solid = $\pi \cdot {r}^{2} \cdot {h}_{2}$

What we have is:

${h}_{1}$ = 33,${h}_{2}$ = 14

$\frac{\pi \cdot {r}^{2} \cdot {h}_{1}}{3}$ + $\pi \cdot {r}^{2} \cdot {h}_{2}$ = $225 \cdot \pi$

$\frac{\pi \cdot {r}^{2} \cdot 33}{3}$ + $\pi \cdot {r}^{2} \cdot 14$ = $225 \cdot \pi$

$\pi \cdot {r}^{2} \cdot 11$ + $\pi \cdot {r}^{2} \cdot 28$ = $225 \cdot \pi$

$39. \pi \cdot {r}^{2}$ = $225 \cdot \pi$

${r}^{2}$ = $\frac{225}{39}$ = $\frac{75}{13}$

Area of the base of the cylinder = $\pi \cdot {r}^{2}$ = $\pi \cdot \frac{75}{13}$ = $\frac{75 \pi}{13}$