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 #21 # and the height of the cylinder is #4 #. If the volume of the solid is #90 pi#, what is the area of the base of the cylinder?

1 Answer
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Mar 22, 2016

#~~ 25.7#

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" = 4# 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" = 21# 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 * (4 + 1/3 xx 21)#

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"/(4 + 1/3 xx 21)#

#= (90pi)/(11)#

#~~ 25.7#

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