Question

The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. If the total surface area is 616 cm², then the volume of the cylinder is
(a) 1078 cm³
(b) 1232 cm³
(c) 1848 cm³
(d) 924 cm³

Answer 1

(a) 1078 cm³

We have:

$$\begin{gathered} \frac{2\mathrm{\pi }rh}{2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2}=\frac{1}{2} \\ \Rightarrow 4\mathrm{\pi }rh=2\mathrm{\pi }rh+2\mathrm{\pi }{r}^2 \\ \Rightarrow 2\mathrm{\pi }rh=2\mathrm{\pi }{r}^2 \\ \Rightarrow \frac{r}{h}=\frac{1}{1} \end{gathered}$$

$$\begin{gathered} \mathrm{Also},2\mathrm{\pi }rh\mathit{}+2\mathrm{\pi }{r}^2=616 \\ \Rightarrow 2\mathrm{\pi }{r}^2+2\mathrm{\pi }{r}^2=616 \\ \Rightarrow 4\mathrm{\pi }{r}^2=616 \\ \Rightarrow \mathrm{\pi }{r}^2=154 \\ \Rightarrow r^2=\frac{154}{\mathrm{\pi }}{\mathrm{cm}}^2 \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{\pi }{r}^{\mathit{2}}h \\ =\mathrm{\pi }\times \frac{154}{\mathrm{\pi }}\times r \\ =154\times \sqrt{\frac{154\times 7}{22}} \\ =154\times 7 \\ =1078{\mathrm{cm}}^3 \end{gathered}$$