Multiple choice

[CAT 1994] A right circular cone, a right circular cylinder and a hemisphere, all have the same radius, and the heights of the cone and cylinder equal to their diameters. Then their volumes are proportional, respectively to

  1. 1:3:1

  2. 2:1:3

  3. 3:2:1

  4. 1:2:3
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A Correct answer
Explanation

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

AI explanation

Let the common radius be r, making the heights of the cone and cylinder 2r. Using the volume formulas, the volume of the cone is (1/3) times pi times r squared times 2r, which equals (2/3) times pi times r cubed. The volume of the cylinder is pi times r squared times 2r, equaling 2 times pi times r cubed, and the volume of the hemisphere is (2/3) times pi times r cubed. Dividing these volumes by (2/3) times pi times r cubed gives the ratio 1:3:1.

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