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 `27` and the height of the cylinder is `7`. If the volume of the solid is `96 pi`, what is the area of the base of the cylinder?

Answer 1

`6pi ~~ 18.850`

Explanation:

The volume of the cylinder is given by its height multiplied by the area of its circular base.

`V_"cylinder" = pi * r^2 * h_"cylinder"`

• `h_"cylinder" = 7` in this question.

The volume of a cone is given by a third of its height multiplied by the area of its circular base.

`V_"cone" = 1/3 * pi * r^2 * h_"cone"`

• `h_"cone" = 27` in this question.
• The variable `r` is reused as the cone has the same radius as the cylinder.

The volume of the entire solid is

`V_"solid" = V_"cylinder" + V_"cone"`

`= pi * r^2 * h_"cylinder" + 1/3 * pi * r^2 * h_"cone"`

`= pi * r^2 * (h_"cylinder" + 1/3 h_"cone")`

`= pi * r^2 * (7 + 1/3 xx 27)`

Now it becomes a simple matter to solve for the base area of the cylinder, which is just `pi r^2`.

`pi * r^2 = V_"solid"/(7 + 1/3 xx 27)`

`= (96pi)/16`

`= 6pi`

`~~ 18.850`