The ratio between the radius of the base and the height of a cylinder is 2 : 3. If its volume is 1617 cm³ the total surface area of the cylinder is :

1. 308 cm²

2. 462 cm²

3. 540 cm²

4. 770 cm²

Let radius of cylinder, r = 2a and height, h = 3a.

Given,

Volume of cylinder = 1617 cm³

By formula,

Volume of cylinder = πr²h

$\Rightarrow 1617 = \dfrac{22}{7} \times (\text{2a})^2 \times \text{3a} \\[1em] \Rightarrow 1617 \times \dfrac{7}{22} = 4(\text{a})^2 \times \text{3a} \\[1em] \Rightarrow \dfrac{11319}{22} = 12\text{a}^3 \\[1em] \Rightarrow \dfrac{514.5}{12} = \text{a}^3 \\[1em] \Rightarrow \text{a}^3 = 42.875 \\[1em] \Rightarrow \text{a} = \sqrt[3]{42.875} \\[1em] \Rightarrow \text{a} = 3.5$

∴ Radius = 2a = 2 × 3.5 = 7 cm

Height = 3a = 3 × 3.5 = 10.5 cm

Total surface area of cylinder = 2πr(r + h)

$= 2 \times \dfrac{22}{7} \times 7 (7 + 10.5) \\[1em] = 2 \times 22 \times 17.5 \\[1em] = 770 \text{cm}^2.$

Hence, option 4 is the correct option.