A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.

Volume of a hemisphere = (2/3)πr^{3}

Volume of a right circular cone = (1/3)πr^{2}h

Volume of a cylinder = πr^{2}h

Given, a cone, a hemisphere and a cylinder stand on equal bases and have the same height.

Height of a hemisphere is the radius and equal bases implies equal base radius.

Thus, height of cone = height of cylinder = base radius = r

Ratio of volumes = (1/3)πr^{2}h : (2/3)πr^{3} : πr^{2}h

⇒ Ratio of volumes = r^{3} : 2r^{3} : 3r^{3} = 1 : 2 : 3

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