A solid cylinder has a total surface area of 231 cm². If its curved surface area is two-thirds of the total surface area, the volume of the cylinder is :

1. 269.5 cm³

2. 308 cm³

3. 363.4 cm³

4. 385 cm³

Given,

Total surface area of cylinder = 231 cm²

⇒ 2πr² + 2πrh = 231 …(1)

Curved surface area = $\dfrac{2}{3}$ (Total surface area)

= $\dfrac{2}{3} \times 231 = 2 \times 77$ = 154 cm²

By formula,

Curved surface area of cylinder = 2πrh

⇒ 2πrh = 154 …(2)

Substituting eq.(2) in eq.(1), we have :

⇒ 2πr² + 154 = 231

⇒ 2πr² = 231 - 154

⇒ 2πr² = 77

$\Rightarrow \text{r}^2 = \dfrac{77}{2π} \\[1em] \Rightarrow \text{r}^2 = \dfrac{77}{2 \times \dfrac{22}{7}} \\[1em] \Rightarrow \text{r}^2 = \dfrac{77 \times 7}{2 \times 22} \\[1em] \Rightarrow \text{r}^2 = \dfrac{539}{44} \\[1em] \Rightarrow \text{r}^2 = 12.25 \\[1em] \Rightarrow \text{r} = \sqrt{12.25} \\[1em] \Rightarrow \text{r} = 3.5 \text{ cm.}$

Substituting value of r in eq.(2), we get:

$\Rightarrow 2 \times \dfrac{22}{7} \times 3.5 \times \text{h} = 154 \\[1em] \Rightarrow 2 \times 22 \times 0.5 \times \text{h} = 154 \\[1em] \Rightarrow 22\text{h} = 154 \\[1em] \Rightarrow \text{h} = \dfrac{154}{22} \\[1em] \Rightarrow \text{h} = 7 \text{ cm.}$

Volume of cylinder = πr²h

$= \dfrac{22}{7} \times 3.5^2 \times 7 \\[1em] = 22 \times 12.25 \\[1em] = 269.5 \text{ cm}^3.$

Hence, option 1 is the correct option.