# 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 60  and the height of the cylinder is 15 . If the volume of the solid is 7 pi, what is the area of the base of the cylinder?

Oct 23, 2016

The area of the base of the cylinder is $\frac{7 \pi}{100}$.

#### Explanation:

The volume of the solid is found by adding the volume of the cylinder to the volume of the cone. The volume of a cylinder is given by $V = \pi {r}^{2} h$. The volume of the cone is given by $V = \frac{\pi {r}^{2} h}{3}$. So the volume of the entire solid is given by:

$V = \pi {r}^{2} h + \frac{\pi {r}^{2} h}{3}$

This can be transformed by the converse of the Distributive Property to become:

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

Which then becomes:

$V = \frac{4 \pi {r}^{2} h}{3}$

Since we are looking for the area of the base, which is a circle, we need to isolate the part of the formula which gives that area. The area of a circle is given by $A = \pi {r}^{2}$, so we need to isolate that part of the simplified volume formula.

$3 \cdot V = 3 \cdot \frac{4 \pi {r}^{2} h}{3}$
$3 V = 4 \pi {r}^{2} h$
$\frac{3 V}{4 h} = \frac{4 \pi {r}^{2} h}{4 h}$
$\frac{3 V}{4 h} = \pi {r}^{2}$

Therefore, the area of the base of the cylinder is given by the formula:

$A = \frac{3 V}{4 h}$

Substitute the values you know for $V$ and $h$ and simplify as possible. {$h = 60 + 15 = 75$}

$A = \frac{3 \cdot 7 \pi}{4 \cdot 75}$
$A = \frac{21 \pi}{300}$
$A = \frac{7 \pi}{100}$

Since the volume in the problem is left in terms of $\pi$, it is acceptable to give the area of the base in terms of $\pi$ as well.