Question

Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.
[CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{radii}{r}_1=6\mathrm{cm},r_2=8\mathrm{cm}\mathrm{and}{r}_3=10\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{resulting}\mathrm{sphere}\mathrm{be}R. \\ \mathrm{As}, \\ \mathrm{Volume}\mathrm{of}\mathrm{resulting}\mathrm{sphere}=\mathrm{Volume}\mathrm{of}\mathrm{three}\mathrm{metallic}\mathrm{spheres} \\ \Rightarrow \frac{4}{3}\mathrm{\pi }{R}^3=\frac{4}{3}\mathrm{\pi }{r_1}^3+\frac{4}{3}\mathrm{\pi }{r_2}^3+\frac{4}{3}\mathrm{\pi }{r_3}^3 \\ \Rightarrow \frac{4}{3}\mathrm{\pi }{R}^3=\frac{4}{3}\mathrm{\pi }\left({r_1}^3+{r_2}^3+{r_3}^3\right) \\ \Rightarrow R^3={r_1}^3+{r_2}^3+{r_3}^3 \\ \Rightarrow R^3=6^3+8^3+{10}^3 \\ \Rightarrow R^3=216+512+1000 \\ \Rightarrow R^3=1728 \\ \Rightarrow R=\sqrt[3]{1728} \\ \Rightarrow R=12\mathrm{cm} \end{gathered}$$

So, the radius of the resulting sphere is 12 cm.