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?

1 Answer
Jul 28, 2016

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