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their volumes are proportional

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by sanju09 » Thu Aug 12, 2010 8:29 pm
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 their diameters. Then their volumes are proportional, respectively to,
(A) 1:3:1
(B) 2:1:3
(C) 3:2:1
(D) 1:2:3
(E) 1:3:2

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by Rahul@gurome » Fri Aug 13, 2010 7:42 pm
Solution:
Let the radius of the cone be r.
So the height of cone and cylinder is 2r.
Volume of a cone is given by (1/3)*pi*r^2*h = (1/3)*pi*r^2*2r = (1/3)*pi*2*r^3 = (2/3)*pi*r^3.
Volume of cylinder is pi*(r^2)*h = pi*2*r^3 = 2*pi*r^3.
Volume of hemisphere is (2/3)*pi*r^3.
So the volumes of cone, cylinder and hemisphere are in the ratio (2/3)*pi*r^3 : 2*pi*r^3 : (2/3)*pi*r^3 = 2/3 : 2 : 2/3 = 1 : 3 : 1.

The correct answer is (A)
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