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 `39` and the height of the cylinder is `17`. If the volume of the solid is `150 pi`, what is the area of the base of the cylinder?

Answer 1

The area of the base of the cylinder is `5pi`.

Explanation:

The formula for volume of a cone is:
`V=pir^2h/3`

The formula for volume of a cylinder is:
`V=pir^2h`

Therefore the formula for the total volume (`V_T`) of the given solid is:

`V_T=pir^2h_1/3+pir^2h_2` (where `h_1=`height of cone, and `h_2=`height of cylinder. The radius, `r`, is the same for both.)

`V_T=pir^2(h_1/3+h_2)`

We need to calculate `r` in order to calculate the area of the base of the cylinder, hence we fill in the data given.

`150pi=pir^2(39/3+17)`

We cancel the like term (`pi`) on each side.

`150cancelpi=cancelpir^2(39/3+17)`

`150=r^2(13+17)`

`150=r^2xx30`

Divide both sides by `30`.

`150/30=r^2`

`5=r^2`

The formula of area of the base of a cylinder (a circle) is:

`A=pir^2`

`A=pixx5`

`A=5pi`