A sphere of radius $10\text{cm}$ is melted and made into a cone of height $10\text{cm}$. Find the diameter of the cone.

When a solid is melted and recast into another shape, its volume remains the same.
$$\text{Volume of Sphere}=\text{Volume of Cone}$$

Let the radius of the sphere be $R=10\text{cm}$, the height of the cone be $h=10\text{cm}$, and the radius of the cone be $r$.

$$\frac{4}{3}\pi R^3=\frac{1}{3}\pi r^{2}h$$

$$4R^3=r^{2}h$$

$$4\times (10{)}^3=r^2\times 10$$

$$4\times 1000=10r^2$$

$$4000=10r^2⟹r^2=400⟹r=20\text{cm}$$

$$\text{Diameter of the cone}=2r=2\times 20=40\text{cm}$$