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 `48 pi`, what is the area of the base of the cylinder?

Answer 1

`3.2pi`

Explanation:

Volume of a right circular cone is given by the expression
`V_"cone"=pir^2h/3`, where `pir^2` is the area of the base and `h` is the vertical height.
Also that Volume of a cylinder `V_"cylinder"=pir^2h`, where `pir^2` is the area of the base and `h` is cylinder's height.

Volume of the given solid `=V_"cone"+V_"cylinder"`
`=1/3"area of the base of cone"xx"height of cone"+"area of the base"xx"height of cylinder"`
Inserting given values

`48pi=1/3"area of the base of cone"xx9+"area of the base of cylinder"xx12`

Since, radius of both cone and cylinder are equal, rearranging and solving for area of the base of cylinder
`15xx"area of the base of cylinder"=48pi`
`"Area of the base of cylinder"=cancel48^16/cancel15_5pi`
`=3.2pi`