Question

A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them being 3.5 cm and the total height of the solid is
9.5 cm. Find the volume of the solid. [CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Radius}\mathrm{of}\mathrm{cone}=\mathrm{Radius}\mathrm{of}\mathrm{hemisphere}=r=3.5\mathrm{cm}\mathrm{or}\mathrm{AD}=\mathrm{BD}=\mathrm{CD}=3.5\mathrm{cm}, \\ \mathrm{Total}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{solid},\mathrm{OC}=9.5\mathrm{cm} \\ \Rightarrow \mathrm{OD}+\mathrm{CD}=9.5 \\ \Rightarrow \mathrm{OD}+3.5=9.5 \\ \Rightarrow \mathrm{OD}=6\mathrm{cm} \\ \Rightarrow \mathrm{Height}\mathrm{of}\mathrm{cone},h=6\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{solid}=\mathrm{Volume}\mathrm{of}\mathrm{cone}+\mathrm{Volume}\mathrm{of}\mathrm{hemisphere} \\ =\frac{1}{3}\mathrm{\pi }{r}^{2}h+\frac{2}{3}\mathrm{\pi }{r}^3 \\ =\frac{1}{3}\mathrm{\pi }{r}^2\left(h+2r\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times \left(6+2\times 3.5\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times \left(6+7\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times 13 \\ =\frac{500.5}{3} \\ \approx 166.83{\mathrm{cm}}^3 \end{gathered}$$

So, the volume of the solid is 166.83 cm³.