Multiple choice

A right circular cone and a cylinder have a circle of unit radius as base and their heights are equal to the radius itself and a hemisphere has the same radius then their volumes are proportional respectively to

  1. 1 : 2 : 3

  2. 3 : 2 : 1

  3. 2 : 1 : 3

  4. 1 : 3 : 2
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D Correct answer
Explanation

Cone: (1/3) * pi * r^2 * h = (1/3) * pi * 1^2 * 1 = pi/3. Cylinder: pi * r^2 * h = pi * 1^2 * 1 = pi. Hemisphere: (2/3) * pi * r^3 = (2/3) * pi * 1^3 = 2pi/3. Ratio: pi/3 : pi : 2pi/3 = 1 : 3 : 2.

AI explanation

Using the unit radius of 1, the volume of the cone is (1/3) * pi * r^2 * h, which equals 1/3 pi. The volume of the cylinder is pi * r^2 * h, which equals pi, and the volume of the hemisphere is (2/3) * pi * r^3, which also equals 2/3 pi. Multiplying all three volumes by 3 to remove the denominators results in the proportion 1 : 3 : 2.

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