Question

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 `9` and the height of the cylinder is `12`. If the volume of the solid is `15 pi`, what is the area of the base of the cylinder?

Answer 1

`pi`

Explanation:

The base of a cylinder is a circle. The area of a circle is `pir^2`; however, we do not know the radius. To find `r`, we can use the information given in the problem.

The volume of a cone is `1/3pir^2h`, where `r` is the radius and `h` is the height. We know the height is `9`, so we can say

`V_"cone" = 1/3pir^2*9 = 3pir^2`

The volume of a cylinder is `pir^2h`, and we know the height is `12`.

`V_"cylinder" = pir^2*12 = 12pir^2`

We also know that the volume of the cone plus that of the cylinder is equal to `15pi`.

`V_"cone" + V_"cylinder" = 15pi`

`3pir^2 + 12pir^2 = 15pi`

`15pir^2 = 15pi`

Dividing by `15pi` on both sides, we get

`r^2 = 1`

`r=1`

So, the area of the base of the cylinder is

`pir^2 = pi* 1^2 = pi`