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

Given:

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

Total surface area of cylinder = 616 cm²

$⇒ \dfrac{\text {Curved surface area of cylinder}}{\text{Total surface area of cylinder}} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{2πrh}{2πr(r + h)} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{\cancel{2πr}h}{\cancel{2πr}(r + h)} = \dfrac{1}{2}\\[1em] ⇒ \dfrac{h}{r + h} = \dfrac{1}{2}\\[1em] ⇒ 2 \times h = 1 \times (r + h)\\[1em] ⇒ 2h = r + h\\[1em] ⇒ 2h - h = r \\[1em] ⇒ h = r$

And,

$⇒ 2πr(r + h) = 616 \\[1em] ⇒ 2 \times \dfrac{22}{7} \times r(r + h) = 616 \\[1em]$

Using r = h

$⇒ 2 \times \dfrac{22}{7} \times r(r + r) = 616 \\[1em] ⇒ \dfrac{44}{7} \times r \times 2r = 616 \\[1em] ⇒ \dfrac{44}{7} \times 2r^2 = 616 \\[1em] ⇒ \dfrac{88}{7} \times r^2 = 616 \\[1em] ⇒ r^2 = \dfrac{7 \times 616}{88} \\[1em] ⇒ r^2 = \dfrac{4,312}{88} \\[1em] ⇒ r^2 = 49 \\[1em] ⇒ r = \sqrt{49} \\[1em] ⇒ r = 7$

And, volume of the cylinder = πr²h

$= \dfrac{22}{7} \times 7^2 \times 7\\[1em] = \dfrac{22}{\cancel{7}} \times 7^2 \times \cancel{7}\\[1em] = 22 \times 49 \\[1em] = 1,078 cm^3$

Hence, the volume of the cylinder is 1,078 cm³.