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Question

A right circular cone and a sphere have equal volumes. If the radius of the base of the cone is 2x and the radius of the sphere is 3x, find the height of the cone in terms of x.

A
x
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B
3x2
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C
4x3
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D
20x
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E
27x
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Solution

The correct option is C 27x
Given, volumes of right circular cone and sphere are equal.
The radius of right circular cone is 2x and sphere is 3x.
Let the height of right circular cone is h.
Then volume of right circular cone = 13πr2h=13π(2x)2h=43πx2h.
Then volume of sphere = 43πr3=43π(3x)3=43×27πx3
But volumes of right circular cone and sphere are equal.
43πx2h=43×27πx3
h=27x

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