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

##### 1 Answer

Mar 22, 2016

#### 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"#

#h_"cylinder" = 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"#

#h_"cone" = 27# in this question.- The variable
#r# 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"#

#= pi * r^2 * h_"cylinder" + 1/3 * pi * r^2 * h_"cone"#

#= pi * r^2 * (h_"cylinder" + 1/3 h_"cone")#

#= pi * r^2 * (7 + 1/3 xx 27)#

Now it becomes a simple matter to solve for the base area of the cylinder, which is just

#pi * r^2 = V_"solid"/(7 + 1/3 xx 27)#

#= (96pi)/16#

#= 6pi#

#~~ 18.850#