Three metallic spheres of radius 3 cm, 4 cm, and 5 cm are melted into a single sphere. Find the radius of the resulting sphere.

Let R be the radius of the resulting sphere.

It is known that the volume of solids remains the same in case of the conversion of solids from one shape into another.

∴Volume of three spheres = Volume of the resulting sphere

$\frac{4}{3}\mathrm{\pi }{\left(3\right)}^3+\frac{4}{3}\mathrm{\pi }{\left(4\right)}^3+\frac{4}{3}\mathrm{\pi }{\left(5\right)}^3=\frac{4}{3}{\mathrm{\pi R}}^3$

$\Rightarrow 27+64+125={\mathrm{R}}^3$

$\Rightarrow {\mathrm{R}}^3=216$

$\Rightarrow \mathrm{R}=6 \mathrm{cm}$

Therefore, the radius of the resulting sphere will be 6 cm.