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
Bio
Mar 22, 2016

#6pi ~~ 18.850#

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"#

  • #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#.

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

#= (96pi)/16#

#= 6pi#

#~~ 18.850#

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