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 42  and the height of the cylinder is 6 . If the volume of the solid is 45 pi, what is the area of the base of the cylinder?

Mar 19, 2018

$2.25 \pi$ units squared

Explanation:

To find the area, we first need to find the radius. We know that the total volume of the cylinder plus the cone is $45 \pi$, which is found by adding the volumes of the cylinder ($6 {r}^{2} \pi$) and the cone ($\frac{42 {r}^{2} \pi}{3}$), where $r$ represents the unknown radius of the cone's and cylinder's bases.

Cylinder's volume: $\pi {r}^{2} \cdot h$

Cone's volume: $\frac{\pi {r}^{2} \cdot h}{3}$

$6 {r}^{2} \pi + \frac{42 {r}^{2} \pi}{3} = 45 \pi$

$\frac{18 {r}^{2} \pi}{3} + \frac{42 {r}^{2} \pi}{3} = 45 \pi$

$\frac{60 {r}^{2} \pi}{3} = 45 \pi$

$60 {r}^{2} \pi = 135 \pi$

$60 {r}^{2} = 135$

${r}^{2} = 2.25$

$r = \pm 1.5 \rightarrow$ Disregard the -1.5, a length cannot be negative

Area is $\pi {r}^{2}$

$\pi \cdot {1.5}^{2}$

$\pi \cdot 2.25$

The answer is $2.25 \pi$