Question

The total surface area of a solid cylinder is 231 cm² and its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder.

Answer 1

Curved surface area = $\frac{2}{3}$$\times$total surface area = $\frac{2}{3}\times 231=2\times 77=154{\mathrm{cm}}^2$

Now, total surface area − curved surface area = $2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2-2\mathrm{\pi }rh$

$$\begin{gathered} \mathrm{Then}231-154=2\mathrm{\pi }{r}^2 \\ \Rightarrow 2\times \frac{22}{7}\times r^2=77 \\ \Rightarrow r^2=\frac{77\times 7}{44}=12.25 \\ \Rightarrow r=3.5\mathrm{cm} \\ \mathrm{Also},\mathrm{curved}\mathrm{surface}\mathrm{area}=154{\mathrm{cm}}^2 \\ \Rightarrow 2\mathrm{\pi }rh=154 \\ \Rightarrow 2\times \frac{22}{7}\times 3.5\times h\mathit{}=154 \\ \Rightarrow h=\frac{154\times 7}{44\times 3.5}=7\mathrm{cm} \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{\pi }{r}^{\mathit{2}}h\mathit{} \\ =\frac{22}{7}\times (3.5{)}^2\times 7=269.5{\mathrm{cm}}^3 \end{gathered}$$