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, the volume of the cylinder is :

1. 1232 cm³

2. 1078 cm³

3. 1848 cm³

4. 1548 cm³

Total surface area = 616 cm²

⇒ 2πr(h + r) = 616

⇒ πr(h + r) = $\dfrac{616}{2}$

⇒ πr(h + r) = 308 ….(1)

Ratio between its curved surface area and total surface area = 1 : 2

$\Rightarrow \dfrac{\text{Curved surface area}}{\text{Total surface area}} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{2π\text{rh}}{2π\text{r(h + r)}} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{\text{h}}{\text{(h + r)}} = \dfrac{1}{2} \\[1em]$

⇒ 2h = h + r

⇒ 2h - h = r

⇒ h = r

Substituting value of h in eq.(1), we get:

⇒ πr(r + r) = 308

⇒ πr × 2r = 308

⇒ 2πr² = 308

$\Rightarrow 2 \times \dfrac{22}{7} \times \text{r}^2 = 308 \\[1em] \Rightarrow \dfrac{44}{7} \times \text{r}^2 = 308 \\[1em] \Rightarrow \text{r}^2 = \dfrac{308 \times 7}{44} \\[1em] \Rightarrow \text{r}^2 = \dfrac{2156}{44} \\[1em] \Rightarrow \text{r}^2 = 49 \\[1em] \Rightarrow \text{r} = \sqrt{49} \\[1em] \Rightarrow \text{r} = 7 \text{ cm.}$

⇒ h = 7 cm

Volume of cylinder = πr²h

= $\dfrac{22}{7}$ × 7² × 7

= 22 × 49

= 1078 cm³.

Hence, option 2 is the correct option.