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.

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 $2\sqrt{\frac{r^3}{h}}$.