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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 $6 \mathrm{~cm}$.