The total surface area of a solid cylinder is 462 cm² and its curved surface area is one-third of its total surface area. Find the volume of the cylinder.

Given,

Total surface area = 462 cm²

⇒ 2πr(h + r) = 462

⇒ πr(h + r) = $\dfrac{462}{2}$

⇒ πr(h + r) = 231 ….(1)

Given, curved surface area is one-third of its total surface area.

⇒ Curved surface area = $\dfrac{1}{3}$ × total surface area

$\Rightarrow \dfrac{\text{curved surface area}}{\text{total surface area}} = \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{2π\text{rh}}{2π\text{r(h + r)}} = \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{\text{h}}{\text{(h + r)}} = \dfrac{1}{3} \\[1em]$

⇒ 3h = h + r

⇒ 3h - h = r

⇒ 2h = r

⇒ h = $\dfrac{\text{r}}{2}$

Substituting value of h in eq.(1), we have:

$\Rightarrow π\text{r(h + r)} = 231 \\[1em] \Rightarrow \dfrac{22}{7} \times \text{r}\Big(\dfrac{\text{r}}{2} + \text{r}\Big) = 231 \\[1em] \Rightarrow \dfrac{22}{7} \times \text{r}\Big(\dfrac{\text{r} + 2\text{r}}{2}\Big) = 231 \\[1em] \Rightarrow \dfrac{22}{7} \times \text{r} \times \dfrac{3\text{r}}{2} = 231 \\[1em] \Rightarrow \dfrac{22}{7} \times \dfrac{3\text{r}^2}{2} = 231 \\[1em] \Rightarrow \text{r}^2 = \dfrac{7 \times 2 \times 231}{22 \times 3} \\[1em] \Rightarrow \text{r}^2 = \dfrac{3234}{66} \\[1em] \Rightarrow \text{r}^2 = 49 \\[1em] \Rightarrow \text{r} = \sqrt{49} \\[1em] \Rightarrow \text{r} = 7 \text{ cm}.$

⇒ h = $\dfrac{\text{r}}{2} = \dfrac{7}{2} = 3.5 \text{ cm}.$

Volume of cylinder = πr²h

$= \dfrac{22}{7} \times 7^2 \times 3.5 \\[1em] = \dfrac{22}{7} \times 49 \times 3.5 \\[1em] = \dfrac{3773}{7} \\[1em] = 539 \text{ cm}^3.$

Hence, volume of cylinder is 539 cm³.