A copper sphere of radius $ 3 \mathrm{~cm} $ is melted and recast into a right circular cone of height 3 $ \mathrm{cm} $. Find the radius of the base of the cone.

Given:

A copper sphere of radius \( 3 \mathrm{~cm} \) is melted and recast into a right circular cone of height 3 \( \mathrm{cm} \).

To do:

We have to find the radius of the base of the cone.

Solution:

Radius of the copper sphere $r=3 \mathrm{~cm}$

Volume of the copper sphere $=\frac{4}{3} \pi r^{3}$

$=\frac{4}{3} \pi \times(3)^{3}$

$=36 \pi \mathrm{cm}^{3}$

Volume of the cone formed from sphere $=$ Volume of the copper sphere 

$=36 \pi \mathrm{cm}^{3}$
Height of the cone $h=3 \mathrm{~cm}$
Let $R$ be the radius of the cone.

Therefore,

$\frac{1}{3} \pi R^{2} h=36 \pi$

$\Rightarrow \frac{1}{3} R^{2}(3)=36 \pi$

$\Rightarrow \pi R^{2}=36 \pi$

$\Rightarrow R^{2}=36$

$\Rightarrow R^{2}=(6)^{2}$

$\Rightarrow R=6\ cm$

The radius of the cone is $6 \mathrm{~cm}$.

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Updated on: 10-Oct-2022

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