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 `42` and the height of the cylinder is `6`. If the volume of the solid is `45 pi`, what is the area of the base of the cylinder?

Answer 1

`2.25pi` units squared

Explanation:

To find the area, we first need to find the radius. We know that the total volume of the cylinder plus the cone is `45pi`, which is found by adding the volumes of the cylinder (`6r^2pi`) and the cone (`(42r^2pi)/3`), where `r` represents the unknown radius of the cone's and cylinder's bases.

Cylinder's volume: `pir^2*h`

Cone's volume: `(pir^2*h)/3`

Add the two volumes together:

`6r^2pi+(42r^2pi)/3=45pi`

`(18r^2pi)/3 +(42r^2pi)/3=45pi`

`(60r^2pi)/3=45pi`

`60r^2pi=135pi`

`60r^2=135`

`r^2=2.25`

`r=+-1.5 rarr` Disregard the -1.5, a length cannot be negative

Area is `pir^2`

`pi*1.5^2`

`pi*2.25`

The answer is `2.25pi`