Question

The ratio of the curved surface area to the total surface area of a right circular cylinder is 1 : 3. Find the volume of the cylinder if its total surface area is 1848 cm².

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylinder}\mathrm{be}r\mathrm{and}\mathrm{its}\mathrm{height}\mathrm{be}h. \\ \mathrm{As}, \\ \frac{\mathrm{CSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}}{\mathrm{TSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}}=\frac{1}{3} \\ \Rightarrow \mathrm{CSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\frac{1}{3}\times \mathrm{TSA}\mathrm{of}\mathrm{the}\mathrm{cylinder} \\ \Rightarrow \mathrm{CSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\frac{1}{3}\times 1848 \\ \Rightarrow \mathrm{CSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}=616{\mathrm{cm}}^2 \\ \Rightarrow 2\mathrm{\pi }rh=616.....\left(\mathrm{i}\right) \\ \mathrm{Also}, \\ \mathrm{TSA}\mathrm{of}\mathrm{the}\mathrm{cylinder}=1848{\mathrm{cm}}^2 \\ \Rightarrow 2\mathrm{\pi rh}+2\mathrm{\pi }{r}^2=1848 \\ \Rightarrow 616+2\mathrm{\pi }{r}^2=1848\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow 2\mathrm{\pi }{r}^2=1848-616 \\ \Rightarrow 2\times \frac{22}{7}\times r^2=1232 \\ \Rightarrow r^2=\frac{1232\times 7}{2\times 22} \\ \Rightarrow r^2=196 \\ \Rightarrow r=\sqrt{196} \\ \Rightarrow r=14\mathrm{cm} \\ \mathrm{Substituting}r=14\mathrm{cm}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get} \\ 2\mathrm{\pi }\times 14\times h=616 \\ \Rightarrow 2\times \frac{22}{7}\times 14\times h=616 \\ \Rightarrow 88h=616 \\ \Rightarrow h=\frac{616}{8} \\ \Rightarrow h=7\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{\pi }{r}^{2}h \\ =\frac{22}{7}\times 14\times 14\times 7 \\ =4312{\mathrm{cm}}^3 \end{gathered}$$

So, the volume of the cylinder is 4312 cm³.