Question

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. [CBSE 2014]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}=\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cylinder}=h=2.8\mathrm{cm}\mathrm{and} \\ \mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{base},r=\frac{4.2}{2}=2.1\mathrm{cm} \\ \mathrm{The}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone},l=\sqrt{r^2+h^2} \\ =\sqrt{2.1^2+2.8^2} \\ =\sqrt{4.41+7.84} \\ =\sqrt{12.25} \\ =3.5\mathrm{cm} \\ \mathrm{Now},\mathrm{the}\mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{remaining}\mathrm{solid}=\mathrm{CSA}\mathrm{of}\mathrm{cylinder}+\mathrm{CSA}\mathrm{of}\mathrm{cone}+\mathrm{Area}\mathrm{of}\mathrm{a}\mathrm{base} \\ =2\mathrm{\pi }rh+\mathrm{\pi }rl+\mathrm{\pi }{r}^2 \\ =\mathrm{\pi }r\left(2h+l+r\right) \\ =\frac{22}{7}\times 2.1\times \left(2\times 2.8+3.5+2.1\right) \\ =22\times 0.3\times \left(5.6+5.6\right) \\ =6.6\times 11.2 \\ =73.92{\mathrm{cm}}^2 \end{gathered}$$

So, the total surface area of the remaining solid is 73.92 cm².