Question

A metallic solid right circular cone is of height 84 cm and the radius of its base is 21 cm. It is melted and recast into a solid sphere. Find the
diameter of the sphere. [CBSE 2014]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cone},h=84\mathrm{cm}\mathrm{and} \\ \mathrm{Base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cone},r=21\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{sphere}\mathrm{be}R. \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{sphere}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{solid}\mathrm{cone} \\ \Rightarrow \frac{4}{3}\mathrm{\pi }{R}^3=\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ \Rightarrow R^3=\frac{r^{2}h}{4} \\ \Rightarrow R^3=\frac{21\times 21\times 84}{4} \\ \Rightarrow R^3=21\times 21\times 21 \\ \Rightarrow R=21\mathrm{cm} \\ \therefore \mathrm{Diameter}=2R=2\times 21=42\mathrm{cm} \end{gathered}$$

So, the diameter of the solid sphere is 42 cm.