Question

The total surface area of a cylinder is 462 cm² and its curved surface area is one-third of its total surface area. Find the volume of the cylinder.

Answer 1

Let r cm be the radii and h cm be the height of the cylinder.

$$\begin{gathered} \mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cylinder}=462{\mathrm{cm}}^2 \\ \mathrm{Curved}\mathrm{surface}\mathrm{area}=\frac{1}{3}\times \mathrm{total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cylinder} \\ \Rightarrow 2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2=462 \\ \mathrm{Also},2\mathrm{\pi }rh=\frac{1}{3}\left(462\right)=154 \\ \mathrm{Then}154+2\mathrm{\pi }{r}^2=462 \\ \Rightarrow 2\mathrm{\pi }{r}^2=308 \\ \Rightarrow r^2=\frac{308\times 7}{2\times 22}=49 \\ \Rightarrow r=7\mathrm{cm} \\ \mathrm{Now},2\mathrm{\pi }\times 7\times h=154 \\ \therefore h=\frac{154}{2\times 22}=3.5\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \mathrm{Hence},\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{\pi }{r}^{\mathit{2}}h \\ =\frac{22}{7}\times 7\times 7\times 3.5 \\ =539{\mathrm{cm}}^3 \end{gathered}$$