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 #1 #. If the volume of the solid is #21 pi#, what is the area of the base of the cylinder?

Answer 1

#3pi ~~ 9.425#

The cylinder's height multiplied by the area of its circular base yields the volume of the cylinder.

#V_"cylinder" = pi * r^2 * h_"cylinder"#

A cone's volume can be calculated by multiplying the area of its circular base by one-third of its height.

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

The total volume of the 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 * (1 + 1/3 xx 18)#
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"/(1 + 1/3 xx 18)#
#= (21pi)/7#
#= 3pi#
#~~ 9.425#
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Answer 2

The area of the base of the cylinder is ( \frac{21\pi}{18} - \frac{1}{3} \times \pi \times 18^2 ). This simplifies to ( \frac{21\pi}{18} - \frac{1}{3} \times \pi \times 324 ), which equals ( \frac{7}{6}\pi - 108\pi ). This further simplifies to ( -\frac{641}{6}\pi ). Thus, the area of the base of the cylinder is ( \frac{641}{6}\pi ) square units.

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