Question

A wooden toy is in the shape of a cone mounted on a cylinder, as shown in the figure. The total height of the toy is 26 cm, while the height of the conical part is 6 cm. The diameter of the base of the conical part is 5 cm and that of the cylindrical part is 4 cm. The conical part and the cylindrical part are respectively painted red and white. Find the area to be painted by each of these colours. [Take $\mathrm{\pi }=\frac{22}{7}$]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{conical}\mathrm{part},r=\frac{5}{2}=2.5\mathrm{cm}, \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{part},R=\frac{4}{2}=2\mathrm{cm}, \\ \mathrm{the}\mathrm{total}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{toy}=26\mathrm{cm}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{conical}\mathrm{part},h=6\mathrm{cm} \\ \mathrm{Also},\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{part},H=26-6=20\mathrm{cm} \\ \mathrm{And},\mathrm{the}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{conical}\mathrm{part},l=\sqrt{r^2+h^2}=\sqrt{2.5^2+6^2}=\sqrt{6.25+36}=\sqrt{42.25}=6.5\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{area}\mathrm{to}\mathrm{be}\mathrm{painted}\mathrm{by}\mathrm{red}\mathrm{colour}=\mathrm{CSA}\mathrm{of}\mathrm{cone}+\mathrm{Area}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{conical}\mathrm{part}-\mathrm{Area}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cylindrical}\mathrm{part} \\ =\mathrm{\pi }rl+\mathrm{\pi }{r}^2-\mathrm{\pi }{R}^2 \\ =\frac{22}{7}\times 2.5\times 6.5+\frac{22}{7}\times 2.5\times 2.5-\frac{22}{7}\times 2\times 2 \\ =\frac{22}{7}\times 16.25+\frac{22}{7}\times 6.25-\frac{22}{7}\times 4 \\ =\frac{22}{7}\times \left(16.25+6.25-4\right) \\ =\frac{22}{7}\times 18.5 \\ \approx 58.14{\mathrm{cm}}^2 \\ \mathrm{Also}, \\ \mathrm{The}\mathrm{area}\mathrm{to}\mathrm{be}\mathrm{painted}\mathrm{by}\mathrm{white}\mathrm{colour}=\mathrm{CSA}\mathrm{of}\mathrm{cylinder}+\mathrm{Area}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cylinder} \\ =2\mathrm{\pi }RH+\mathrm{\pi }{R}^2 \\ =\mathrm{\pi }R\left(2H+R\right) \\ =\frac{22}{7}\times 2\times \left(2\times 20+2\right) \\ =\frac{22}{7}\times 2\times 42 \\ =264{\mathrm{cm}}^2 \end{gathered}$$