Question

A hemisphere of lead of radius 6 cm is cast into a right circular cone of height 75 cm. Find the radius of the base of the cone.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{hemisphere},R=6\mathrm{cm}\mathrm{and} \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{cone},h=75\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{base}\mathrm{of}\mathrm{the}\mathrm{cone}\mathrm{be}r. \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{hemisphere} \\ \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{2}h=\frac{2}{3}\mathrm{\pi }{R}^3 \\ \Rightarrow r^2=\frac{2R^3}{h} \\ \Rightarrow r^2=\frac{2\times 6\times 6\times 6}{75} \\ \Rightarrow r^2=5.76 \\ \Rightarrow r=\sqrt{5.76} \\ \therefore r=2.4\mathrm{cm} \end{gathered}$$

So, the radius of the base of the cone is 2.4 cm.