The total surface area of a solid cylinder is 616 cm². If the ratio between its curved surface area and total surface area is 1 : 2; find the volume of the cylinder.

Given,

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

$\Rightarrow \dfrac{\text{Curved surface area}}{\text{Total surface area}} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{2πrh}{2πr(h + r)} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{h}{h + r} = \dfrac{1}{2} \\[1em] \Rightarrow 2h = h + r \\[1em] \Rightarrow 2h - h = r \\[1em] \Rightarrow h = r.$

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

⇒ πr(r + r) = 308

⇒ πr.2r = 308

⇒ 2πr² = 308

⇒ πr² = 154

⇒ $\dfrac{22}{7} \times r^2$ = 154

⇒ r² = $\dfrac{154 \times 7}{22}$

⇒ r² = 49

⇒ r = $\sqrt{49}$

⇒ r = 7 cm

⇒ h = 7 cm.

Volume of cylinder = πr²h

= $\dfrac{22}{7} \times (7)^2 \times 7$

= 22 × 49

= 1078 cm³.

Hence, volume of cylinder = 1078 cm³.