The ratio between the curved surface and the total surface of a cylinder is 1 : 2. Find the volume of the cylinder, given that its total surface area is 616 cm².

We know that, Total surface area = 2πr(h + r) and Curved surface area = 2πrh.

Given ratio between the curved surface and the total surface of a cylinder is 1 : 2.

$\therefore \dfrac{2πrh}{2πr(h + r)} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{h}{h + r} = \dfrac{1}{2} \\[1em] \Rightarrow 2h = h + r \\[1em] \Rightarrow 2h - h = r \\[1em] \Rightarrow h = r.$

Given, total surface area = 616 cm².

∴ 2πr(h + r) = 616

and h = r.

$\Rightarrow 2πr(r + r) = 616 \\[1em] \Rightarrow 2πr.2r = 616 \\[1em] \Rightarrow 4πr^2 = 616 \\[1em] \Rightarrow 4 \times \dfrac{22}{7} \times r^2 = 616 \\[1em] \Rightarrow r^2 = \dfrac{616 \times 7}{22 \times 4} \\[1em] \Rightarrow r^2 = \dfrac{4312}{88} \\[1em] \Rightarrow r^2 = 49 \\[1em] \Rightarrow r = \sqrt{49} = 7 \text{ cm}.$

Since h = r, hence h = 7 cm.

Volume of cylinder = πr²h.

Putting values we get,

$\text{Volume of cylinder } = \dfrac{22}{7}\times (7)^2 \times 7 \\[1em] = \dfrac{22 \times 49 \times 7}{7} \\[1em] = \dfrac{7546}{7} \\[1em] = 1078 \text{ cm}^3.$

Hence, the volume of cylinder = 1078 cm³.