# 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 42  and the height of the cylinder is 10 . If the volume of the solid is 225 pi, what is the area of the base of the cylinder?

Dec 19, 2016

$29.452 c {m}^{2}$

#### Explanation:

Lets assume we work in centimeters.
Top part of solid= a cone,bottom part a cilinder.
$= \frac{1}{3} \pi {r}^{2} h$ = volume cone
$\pi {r}^{2} h$= vol cilinder
$\frac{1}{3} \pi {r}^{2} h + \pi {r}^{2} h = 225 \pi$
multiply both sides with$\frac{1}{\pi}$
${r}^{2} \left(\frac{1}{3} h + h\right) = 225$
${r}^{2} \left(\frac{1}{3} \cdot 42 + 10\right) = 225$
$24 {r}^{2} = 225$
${r}^{2} = 9.375$
substitute ${r}^{2} = 9.375$
$\frac{1}{3} \pi {r}^{2} h + \pi {r}^{2} h = 225 \pi$
$\frac{1}{3} \cdot \pi \cdot 9.375 \cdot 42 + \pi \cdot 9.375 \cdot 10 = 225 \pi$
$412.334 + 294.524 = 706.858$
$706.858 = 707.858$
Area base of cilinder=$\pi {r}^{2}$
$= 3.141592654 \cdot 9.375$
Area of base$= 29.452 c {m}^{2}$

Dec 25, 2016

$A = \frac{75 \pi}{8}$

$A = 29.45$

#### Explanation:

The volume of the cylinder and the cone together is $225 \pi$

Let $h$ = height of cylinder and $H$ = height of cone.

Using the formulae gives:

$\pi {r}^{2} h + \frac{1}{3} \pi {r}^{2} H = 225 \pi$

$\pi {r}^{2} \left(10\right) + \frac{1}{\cancel{3}} \pi {r}^{2} \left({\cancel{42}}^{14}\right) = 225 \pi$

Note that we are not asked for radius, just for the area of the base of the cylinder which is given by $A = \pi {r}^{2}$

Solve for $\pi {r}^{2}$

$10 \pi {r}^{2} + 14 \pi {r}^{2} = 225 \pi$

$24 \pi {r}^{2} = 225 \pi$

$\pi {r}^{2} = \frac{225 \pi}{24}$

$A = \frac{75 \pi}{8}$