# 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 66  and the height of the cylinder is 5 . If the volume of the solid is 64 pi, what is the area of the base of the cylinder?

Jun 6, 2018

$A = \frac{64}{27} \pi$ ${u}^{2}$

#### Explanation:

The volume of the cone is given by: $v = \frac{1}{3} \pi {r}^{2} h$
Since the height of the cone is 66, then $h = 66$
So, $v = \frac{1}{3} \pi {r}^{2} \times 66 = 22 \pi {r}^{2}$

The volume of a cylinder is given by: $v = \pi {r}^{2} h$
Since the height of the cylinder is 5, then $h = 5$
So, $v = \pi {r}^{2} h = \pi {r}^{2} \times 5 = 5 \pi {r}^{2}$

The total volume of the solid is $64 \pi$
Therefore, $22 \pi {r}^{2} + 5 \pi {r}^{2} = 64 \pi$

$27 \pi {r}^{2} = 64 \pi$
${r}^{2} = \frac{64 \pi}{27 \pi}$
${r}^{2} = \frac{64}{27}$
$r = \pm \frac{8}{3 \sqrt{3}}$
Since r is the radius, it must be have the restriction: $r > 0$
Therefore, $r = \frac{8}{3 \sqrt{3}}$units

To find the base of the cylinder, we need to know that the base is a circle. The area of a circle is given by $A = \pi {r}^{2} = \pi \times {\left(\frac{8}{3 \sqrt{3}}\right)}^{2} = \frac{64}{27} \pi$ ${u}^{2}$

Jun 6, 2018

The area of the base of the cylinder is: $A = \pi {r}^{2} = \frac{64 \pi}{27}$

#### Explanation:

The area of the base we need to find is: $A = \pi {r}^{2}$, where $r$ is the radius of the cylinder.

The volume of the cylinder is: $\pi {r}^{2} \cdot {h}_{1}$
where ${h}_{1}$ is the height of the cylinder.
The volume of the cone is $\pi {r}^{2} \cdot {h}_{2} / 3$
where ${h}_{2}$ is the height of the cone.

The volume of the solid is the sum of those two volumes: $V = \pi {r}^{2} \cdot {h}_{1} + \pi {r}^{2} \cdot {h}_{2} / 3$
Factoring $\pi {r}^{2}$:
$V = \pi {r}^{2} \left({h}_{1} + {h}_{2} / 3\right)$
$64 \pi = \pi {r}^{2} \left(5 + \frac{66}{3}\right) = \pi {r}^{2} \left(5 + 22\right) = \pi {r}^{2} \left(27\right)$

$64 \pi = 27 \pi {r}^{2}$

$\pi {r}^{2} = \frac{64 \pi}{27}$
And that is the area of the base: $A = \pi {r}^{2} = \frac{64 \pi}{27}$