The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2.

(a) 1078 cm3

We have:

$\frac{2 \pi r h}{2 \pi r h+2 \pi r^{2}}=\frac{1}{2}$

$\Rightarrow 4 \pi r h=2 \pi r h+2 \pi r^{2}$

$\Rightarrow 2 \pi r h=2 \pi r^{2}$

$\Rightarrow \frac{r}{h}=\frac{1}{1}$

Also, $2 \pi r h+2 \pi r^{2}=616$

$\Rightarrow 2 \pi r^{2}+2 \pi r^{2}=616$

$\Rightarrow 4 \pi r^{2}=616$

$\Rightarrow \pi r^{2}=154$

$\Rightarrow r^{2}=\frac{154}{\pi} \mathrm{cm}^{2}$

$\therefore$ Volume of the cylinder $=\pi r^{2} h$

$=\pi \times \frac{154}{\pi} \times r$

$=154 \times \sqrt{\frac{154 \times 7}{22}}$

$=154 \times 7$

$=1078 \mathrm{~cm}^{3}$