# 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 420 pi, what is the area of the base of the cylinder?

Jul 30, 2018

color(green)("radius of the cone " = r = sqrt 10, " units"

#### Explanation:

$\text{Given } {h}_{1} = 18 , {h}_{2} = 36 , V = 420 \pi$

$\text{Volume of the solid } = V = {V}_{c o \ne} + {V}_{c y l}$

${V}_{c o \ne} = \frac{1}{3} \pi {r}^{2} h = \frac{1}{3} \pi {r}^{2} 18 = 6 \pi {r}^{2}$

${V}_{c y l} = \pi {r}^{2} h = \pi {r}^{2} 36 = 36 \pi {r}^{2}$

$V = 6 \pi {r}^{2} + 36 \pi {r}^{2} = 420 \pi$

$42 \pi {r}^{2} = 420 \pi$

${r}^{2} = {\cancel{420 \pi}}^{\textcolor{red}{10}} / \cancel{42 \pi} = 10$

color(green)("radius of the cone " = r = sqrt 10, " units"