Question

Three solid metallic spheres of radii 3 cm, 4 cm and 5 cm, respectively, are melted to form a single solid sphere. Find the diameter of the resulting sphere.

Answer 1

Let the radius of the new sphere formed be $r$.
The volume of this new sphere will be equal to the sum of the volumes of the smaller spheres that were melted.
Therefore,

$$\begin{gathered} \frac{4}{3}\pi r^3=\frac{4}{3}\pi {\left(3\right)}^3+\frac{4}{3}\pi {\left(4\right)}^3+\frac{4}{3}\pi {\left(5\right)}^3 \\ \Rightarrow \frac{4}{3}\pi r^3=\frac{4}{3}\pi \left(3^3+4^3+5^3\right) \\ \text{O}r, \\ {r}^3=216 \\ \Rightarrow r=\sqrt[3]{216}=6\mathrm{cm} \\ \Rightarrow d=2\times 6\mathrm{cm}=12\mathrm{cm} \end{gathered}$$

The diameter of the resulting sphere is 12 cm.