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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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