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 `18` and the height of the cylinder is `21`. If the volume of the solid is `20 pi`, what is the area of the base of the cylinder?

Answer 1

`A_(base) = pi r^2 = 20/27 pi ~~ 2.327 " units"^2`

Explanation:

Given: A solid with a cone on top of a cylinder. `V_(solid) = 20 pi`
`" "h_("cone") = 18; h_("cylinder") = 21; r = r_("cone") = r_("cylinder")`

Equation for the volume of the solid:

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

`V_("cylinder") = pi r^2 h_("cylinder"); " "V_("cone") = 1/3 pi r^2 h_("cylinder")`

Substitute into the equation the volume of each solid section:
`V_(solid) = pi r^2 h_("cylinder") + 1/3 pi r^2 h_("cylinder")`

Substitute into the equation the known heights:
`20 pi = 21 pi r^2 + 1/3 * 18 pi r^2`

`20 pi = 21 pi r^2 + 6 pi r^2 = 27 pi r^2`

Divide both sides by `pi: " "20 = 27 r^2`

Solve for `r^2: " "20/27 = r^2`

Calculate the area of the base of the cylinder:

`A_base = pi r^2 = 20/27 pi ~~ 2.327 " units"^2`