# 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 15 . If the volume of the solid is 48 pi, what is the area of the base of the cylinder?

Jan 4, 2018

See a solution process below:

#### Explanation:

The formula for the volume of a cylinder is:

${V}_{c} = \pi {r}^{2} {h}_{c}$

The formula for the volume of a cone is:

${V}_{n} = \pi {r}^{2} {h}_{n} / 3$

We can now write an equation for the volume of the complete solid as:

${V}_{c} + {V}_{n} = 48 \pi$

$\pi {r}^{2} {h}_{c} + \pi {r}^{2} {h}_{n} / 3 = 48 \pi$

Where:

• $r$ is the radius of the cone and the cylinder.
• ${h}_{c}$ is the height of the cylinder
• ${h}_{n}$ is the height of the cone

We can substitute what we know from the problem and solve for $r$:

$\left(\pi {r}^{2} \times 15\right) + \left(\pi {r}^{2} \times \frac{9}{3}\right) = 48 \pi$

$15 \pi {r}^{2} + 3 \pi {r}^{2} = 48 \pi$

$\left(15 + 3\right) \pi {r}^{2} = 48 \pi$

$18 \pi {r}^{2} = 48 \pi$

$\frac{18 \pi {r}^{2}}{\textcolor{red}{18} \textcolor{red}{\pi}} = \frac{48 \pi}{\textcolor{red}{18} \textcolor{red}{\pi}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{18 \pi}}} {r}^{2}}{\cancel{\textcolor{red}{18} \textcolor{red}{\pi}}} = \frac{48 \textcolor{red}{\cancel{\textcolor{b l a c k}{\pi}}}}{\textcolor{red}{18} \cancel{\textcolor{red}{\pi}}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{18 \pi}}} {r}^{2}}{\cancel{\textcolor{red}{18} \textcolor{red}{\pi}}} = \frac{48 \textcolor{red}{\cancel{\textcolor{b l a c k}{\pi}}}}{\textcolor{red}{18} \cancel{\textcolor{red}{\pi}}}$

${r}^{2} = \frac{48}{\textcolor{red}{18}}$

${r}^{2} = \frac{6 \times 8}{\textcolor{red}{6 \times 3}}$

${r}^{2} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}} \times 8}{\textcolor{red}{\textcolor{b l a c k}{\cancel{\textcolor{red}{6}}} \times 3}}$

${r}^{2} = \frac{8}{3}$

$\sqrt{{r}^{2}} = \sqrt{\frac{8}{3}}$

$r = \sqrt{\frac{8}{3}}$

The base of a cylinder is a circle. The formula for the area of a circle is:

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

We can substitute the radius we calculate above to determine the area of the circle which is the same as the area of the base of the cylinder:

$A = \pi {\left(\sqrt{\frac{8}{3}}\right)}^{2}$

$A = \pi \frac{8}{3}$

$A = \frac{8}{3} \pi$