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If a sphere of radius $r$ is melted and recasted into a cone of height $h$, then find the radius of the base of the cone.
Given: A sphere of radius $r$ is melted and recasted into a cone of height $h$.
To do: To find the radius of the base of the cone.
Solution:
As given, sphere of radius $r$ is melted and recast into a cone of height $h$. Let $R$ be the radius of the newly formed cone.
Therefore,
Volume of sphere $=\frac{4}{3}\pi r^3$
Volume of cone $=\frac{1}{3}\pi R^2h$
As we know, Volume of sphere$=$Volume of cone
$\Rightarrow \frac{4}{3}\pi r^3=\frac{1}{3} \pi R^2h$
$\Rightarrow 4r^3=R^2h$
$\Rightarrow R^2=\frac{4r^3}{h}$
$\Rightarrow R=\sqrt{\frac{4r^3}{h}}$
$\Rightarrow R=2\sqrt{\frac{r^3}{h}}$
Thus, the radius of thhe cone is $2\sqrt{\frac{r^3}{h}}$.
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