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Question

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


Solution

Let the radius of cone, hemisphere and cylinder be r. Then, height of the hemisphere = r, height of the cone = r, and height of the cylinder = r. Let $$ V_{1},V_{2},V_{3} $$ be the volumes of cone, hemisphere and the cylinder respectively. Then, 
$$ V_{1} = \dfrac{1}{3}\pi r^{2}\times r = \dfrac{1}{3}\pi r^{3}$$

$$V_{2} = \dfrac{2}{3}\pi r^{3} $$ 

$$ V_{3} = \pi r^{2}\times r = \pi r^{3} $$

$$ \therefore V_{1}:V_{2}:V_{3} = 1:2:3 $$

Mathematics

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