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 27 and the height of the cylinder is 7 7. If the volume of the solid is 96 pi96π, what is the area of the base of the cylinder?

1 Answer
Mar 22, 2016

6pi ~~ 18.8506π18.850

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"Vcylinder=πr2hcylinder

  • h_"cylinder" = 7hcylinder=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"Vcone=13πr2hcone

  • h_"cone" = 27hcone=27 in this question.
  • The variable rr 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"Vsolid=Vcylinder+Vcone

= pi * r^2 * h_"cylinder" + 1/3 * pi * r^2 * h_"cone"=πr2hcylinder+13πr2hcone

= pi * r^2 * (h_"cylinder" + 1/3 h_"cone")=πr2(hcylinder+13hcone)

= pi * r^2 * (7 + 1/3 xx 27)=πr2(7+13×27)

Now it becomes a simple matter to solve for the base area of the cylinder, which is just pi r^2πr2.

pi * r^2 = V_"solid"/(7 + 1/3 xx 27)πr2=Vsolid7+13×27

= (96pi)/16=96π16

= 6pi=6π

~~ 18.85018.850