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 $36$ $36$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $126 π$ $126 pi$ , what is the area of the base of the cylinder?

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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 $36$ $36$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $126 π$ $126 pi$ , what is the area of the base of the cylinder?

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Answer 1 · 2016-03-22T07:39:29

$7 π \approx 21.991$ $7pi ~~ 21.991$

Explanation:

The volume of the cylinder is given by its height multiplied by the area of its circular base.

$V_{cylinder} = π \cdot r^{2} \cdot h_{cylinder}$ $V_"cylinder" = pi * r^2 * h_"cylinder"$

$h_{cylinder} = 6$ $h_"cylinder" = 6$ in this question.

The volume of a cone is given by a third of its height multiplied by the area of its circular base.

$V_{cone} = \frac{1}{3} \cdot π \cdot r^{2} \cdot h_{cone}$ $V_"cone" = 1/3 * pi * r^2 * h_"cone"$

$h_{cone} = 36$ $h_"cone" = 36$ in this question.
The variable $r$ $r$ is reused as the cone has the same radius as the cylinder.

The volume of the entire solid is

$V_{solid} = V_{cylinder} + V_{cone}$ $V_"solid" = V_"cylinder" + V_"cone"$

$= π \cdot r^{2} \cdot h_{cylinder} + \frac{1}{3} \cdot π \cdot r^{2} \cdot h_{cone}$ $= pi * r^2 * h_"cylinder" + 1/3 * pi * r^2 * h_"cone"$

$= π \cdot r^{2} \cdot (h_{cylinder} + \frac{1}{3} h_{cone})$ $= pi * r^2 * (h_"cylinder" + 1/3 h_"cone")$

$= π \cdot r^{2} \cdot (6 + \frac{1}{3} \times 36)$ $= pi * r^2 * (6 + 1/3 xx 36)$

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

$π \cdot r^{2} = \frac{V_{solid}}{6 + \frac{1}{3} \times 36}$ $pi * r^2 = V_"solid"/(6 + 1/3 xx 36)$

$= \frac{126 π}{18}$ $= (126pi)/18$

$= 7 π$ $=7pi$

$\approx 21.991$ $~~ 21.991$

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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 $36$ $36$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $126 π$ $126 pi$ , what is the area of the base of the cylinder?

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Explanation:

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Explanation:

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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 $36$ $36$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $126 π$ $126 pi$ , what is the area of the base of the cylinder?