# 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 11 . If the volume of the solid is 140 pi, what is the area of the base of the cylinder?

Mar 23, 2016

The formula for volume of a cone is $\frac{1}{3} {A}_{b a s e} \times h$ while the formula for volume of a cylinder is ${A}_{b a s e} \times h$. The area of the base is the formula for the area of a circle, so $\pi \times {r}^{2}$

#### Explanation:

We are searching for area of base. First though, we must find the radius, r.

$\frac{1}{3} \times 9 \times \left({r}^{2} \times \pi\right) + \left({r}^{2} \times \pi\right) \times 11 = 140 \pi$

$3 \pi {r}^{2} + 11 \pi {r}^{2} = 140 \pi$

$\pi \left(3 {r}^{2} + 11 {r}^{2}\right) = 140 \pi$

$14 {r}^{2} = \frac{140 \pi}{\pi}$

${r}^{2} = 10$

$r = \sqrt{10}$

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

$A = {\left(\sqrt{10}\right)}^{2} \times \pi$

$A = 10 \pi$

The area of the base of the cylinder measures $10 \pi$ square units.

Hopefully this helps!