A cone and a hemisphere have equal bases and equal volumes. Find the ratio Of their heights.

 Given:

A cone and a hemisphere have equal bases and equal volumes. 

To do:

We have to find the ratio of their heights.

Solution:

Let $r$ be the radius of the cone and hemisphere and $h$ be the height of the cone.

This implies,

Volume of the cone $=\frac{1}{3} \pi r^{2} h$

Volume of the hemisphere $=\frac{2}{3} \pi r^{3}$

The volumes of the cone and hemisphere are equal.

Therefore,

$\frac{1}{3} \pi r^{2} h=\frac{2}{3} \pi r^{3}$

$r^{2} h=2 r^{3}$

$h=2 r$

$\Rightarrow \frac{h}{r}=\frac{2}{1}$

The ratio of their heights is $2: 1$.

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Updated on: 10-Oct-2022

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