Three solid metallic spherical balls of radii 3 𝑐𝑚, 4 𝑐𝑚 and 5 𝑐𝑚 are melted to form a single spherical ball, find its radius.

We are given three metallic balls of radii 3 𝑐𝑚, 4 𝑐𝑚 and 5 𝑐𝑚, which are melted to form a single spherical ball. Next, we have to find the radius of the bigger spherical ball. The sum of the volume of three spherical balls will be equal to the volume of a single spherical ball because the volume of a solid is the amount of three-dimensional space occupied by the solid. Whatever the shape of the solid, it will occupy the same amount of space.

Now, the volume of the sphere  = $\frac{1}{3}\mathrm{\pi }\times {\mathrm{r}}^3$, where 𝑟 is the sphere’s radius

According to the given condition,

The volume of a single spherical ball = the sum of the volume of three spherical balls Let the radius of a single spherical ball be 𝑟 𝑐𝑚

$$\begin{gathered} \frac{1}{3}\mathrm{\pi }\times {\mathrm{r}}^3 = \frac{1}{3}\mathrm{\pi }\times 3^3 + \frac{1}{3}\mathrm{\pi }\times 4^3 + \frac{1}{3}\mathrm{\pi }\times 5^3 \\ {\mathrm{r}}^{3 }=3^3 + 4^3 + 5^3 \\ {\mathrm{r}}^3 = 216 \\ \mathrm{r} = \sqrt[3]{216} \\ \mathrm{r} = 6 \end{gathered}$$

Hence, the radius of the single spherical ball is 6 𝑐𝑚