# 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 33  and the height of the cylinder is 13 . If the volume of the solid is 256 pi, what is the area of the base of the cylinder?

Dec 2, 2016

$33.52$

#### Explanation:

A Conical Volume is given by:
$V = \frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot h$
A Cylindrical Volume is given by:
$V = \pi \cdot {r}^{2} \cdot h$
Circular Area (base of cylinder)
$A = 2 \cdot \pi \cdot {r}^{2}$
Total solid volume =
$256 \pi = \frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot {h}_{1} + \pi \cdot {r}^{2} \cdot {h}_{2}$ Solve for r.
$256 = \frac{1}{3} \cdot {r}^{2} \cdot 33 + {r}^{2} \cdot 13$ ; $256 = {r}^{2} \cdot 11 + {r}^{2} \cdot 13$
$256 = {r}^{2} \cdot 24$ ; $\frac{256}{24} = {r}^{2}$ ; r^2 = 10.67 ;

$A = \pi \cdot 10.76 \approx 33.52$

Dec 2, 2016

$10.76 \pi$

#### Explanation:

Let's consider the diagram

We need to find the area of the base of the cylinder, which is a circle. The area of a circle is given by

color(blue)("Area of circle"=pir^2

Where $r$ is the radius of the circle. We need to find ${r}^{2}$

$\approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx$

The total volume of the solid is $256 \pi$

Therefore,

color(purple)("Volume of cone"+"Volume of cylinder"=256pi

We use the formulas

color(orange)("Volume of cone"=1/3pir^2

color(orange)("Volume of cylinder"=pir^2h

Where, $h$ is the height and $r$ is the radius. Let's put everything in the equation,

$\rightarrow \frac{1}{3} \pi {r}^{2} h + \pi {r}^{2} h = 256 \pi$

$\rightarrow \frac{1}{3} \pi {r}^{2} \cdot 33 + \pi {r}^{2} \cdot 13 = 256 \pi$

$\rightarrow \frac{1}{{\cancel{3}}^{1}} \pi {r}^{2} \cdot {\cancel{33}}^{11} + \pi {r}^{2} \cdot 13 = 256 \pi$

$\rightarrow \pi {r}^{2} \cdot 11 + \pi {r}^{2} \cdot 13 = 256 \pi$

$\rightarrow 24 \pi {r}^{2} = 256 \pi$

$\rightarrow 24 \cancel{\pi} {r}^{2} = 256 \cancel{\pi}$

$\rightarrow 24 {r}^{2} = 256$

$\rightarrow {r}^{2} = \frac{256}{24}$

color(violet)(rArrr^2=10.56

$\approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx$

Now, Let's find the area of the base

color(green)("Area"=pir^2=pi*10.76~~33.52