# 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 21 . If the volume of the solid is 20 pi, what is the area of the base of the cylinder?

May 27, 2018

${A}_{b a s e} = \pi {r}^{2} = \frac{20}{27} \pi \approx 2.327 {\text{ units}}^{2}$

#### Explanation:

Given: A solid with a cone on top of a cylinder. ${V}_{s o l i d} = 20 \pi$
" "h_("cone") = 18; h_("cylinder") = 21; r = r_("cone") = r_("cylinder")

Equation for the volume of the solid:

${V}_{s o l i d} = {V}_{\text{cylinder") + V_("cone}}$

${V}_{\text{cylinder") = pi r^2 h_("cylinder"); " "V_("cone") = 1/3 pi r^2 h_("cylinder}}$

Substitute into the equation the volume of each solid section:
${V}_{s o l i d} = \pi {r}^{2} {h}_{\text{cylinder") + 1/3 pi r^2 h_("cylinder}}$

Substitute into the equation the known heights:
$20 \pi = 21 \pi {r}^{2} + \frac{1}{3} \cdot 18 \pi {r}^{2}$

$20 \pi = 21 \pi {r}^{2} + 6 \pi {r}^{2} = 27 \pi {r}^{2}$

Divide both sides by $\pi : \text{ } 20 = 27 {r}^{2}$

Solve for ${r}^{2} : \text{ } \frac{20}{27} = {r}^{2}$

Calculate the area of the base of the cylinder:

${A}_{b} a s e = \pi {r}^{2} = \frac{20}{27} \pi \approx 2.327 {\text{ units}}^{2}$