# 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 18  and the height of the cylinder is 36 . If the volume of the solid is 520 pi, what is the area of the base of the cylinder?

Jun 28, 2016

Here is a diagram.

#### Explanation:

First, let's identify the formulae for volume of a cone and volume of a cylinder.

color(blue)(V_"(cone)" = 1/3pir^2h"

color(red)(V_"(cylinder)" = r^2pi xx h

The total volume of the solid is given by adding the volumes of these two solids together. Therefore, we can state that:

${V}_{\text{total" = h_"(cylinder)"r^2pi + 1/3pir^2h_"(cone)}}$

Let's now identify the elements that we know:

•Total volume
•Height of the cylinder
•Height of the Cone

All that is left for us to find is the radius.

Hence,

$520 \pi = 36 \pi {r}^{2} + \frac{1}{3} \left(18\right) \pi {r}^{2}$

$520 \pi = 36 \pi {r}^{2} + 6 \pi {r}^{2}$

$520 \pi = 42 \pi {r}^{2}$

$\frac{520 \pi}{42 \pi} = {r}^{2}$

$\frac{260}{21} = {r}^{2}$

$\sqrt{\frac{260}{21}} = r$

We can now find the area of the base of the cylinder, which because it's a circle, is given by $a = {r}^{2} \times \pi$.

$a = {\left(\sqrt{\frac{260}{21}}\right)}^{2} \times \pi$

$a = \frac{260 \pi}{21}$

The area of the base of the cylinder is (260pi)/21" u^2.

Hopefully this helps!