# 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 17 . If the volume of the solid is 168 pi, what is the area of the base of the cylinder?

Apr 10, 2016

$6 \pi$

#### Explanation:

Consider the diagram

Note: color(brown)(pi=22/7,V=volume,h=height

Remember the formulas

color(purple)(V_(con)=1/3pir^2h

color(purple)(V_(cyl)=pir^2h

color(purple)(Area_(base)=pir^2

And $r$ is the radius of the base (circle)

We know that the volume of the whole solid is $168 \pi$

Our aim is to find Area of base ($\pi {r}^{2}$)

And we could see that $\pi {r}^{2}$ is present in both of the volume formulas

So, consider $\pi {r}^{2}$ as $a$

$\rightarrow \frac{1}{3} \cdot a \cdot 33 + a \cdot 17 = 168 \pi$

$\rightarrow \frac{1}{\cancel{3}} ^ 1 \cdot a \cdot {\cancel{33}}^{11} + a \cdot 17 = 168 \pi$

$\rightarrow a \cdot 33 + a \cdot 17 = 168 \pi$

$\rightarrow 28 a = 168 \pi$

$\rightarrow a = \frac{168}{28} \pi$

color(green)(rArra=6pi

Source of image: paint (my drawing)

And,if you don't like this drawing and want to change it, your changes are welcome