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 `10`. If the volume of the solid is `144 pi`, what is the area of the base of the cylinder?

Answer 1

`A_("base") = 6pi`

Explanation:

The volume of each component is given by

`V_("cone") = 1/3pir_("cone")^2h_("cone")`

`V_("cylinder") = pir_("cylinder")^2h_("cylinder")`

We have that `r_("cone") = r_("cylinder")` so we shall just denote these as `r`.

`V_("total") = V_("cone") + V_("cylinder")`

`V_("total") = pir^2(1/3h_("cone") + h_("cylinder"))`

`therefore pir^2 = (V_("total"))/(1/3h_("cone") + h_("cylinder"))`

Notice that `pir^2` is precisely the area of the base of the cylinder, which is what we want to calculate so just plug in the numbers:

`A_(base) = (144pi)/(1/3*42 + 10) = (144pi)/(14+10) = (144pi)/(24) = 6pi`