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 42 and the height of the cylinder is 10 10. If the volume of the solid is 144 pi144π, what is the area of the base of the cylinder?

1 Answer
Jul 28, 2016

A_("base") = 6piAbase=6π

Explanation:

The volume of each component is given by

V_("cone") = 1/3pir_("cone")^2h_("cone")Vcone=13πr2conehcone

V_("cylinder") = pir_("cylinder")^2h_("cylinder")Vcylinder=πr2cylinderhcylinder

We have that r_("cone") = r_("cylinder")rcone=rcylinder so we shall just denote these as rr.

V_("total") = V_("cone") + V_("cylinder") Vtotal=Vcone+Vcylinder

V_("total") = pir^2(1/3h_("cone") + h_("cylinder"))Vtotal=πr2(13hcone+hcylinder)

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