The sum of the radius of the base and the height of a solid cylinder is 37 m. If the total surface area of the cylinder be 1628 m², find its volume.

Given, r + h = 37 m

Total surface area = 1628 m²

⇒ 2πr(h + r) = 1628

⇒ πr(h + r) = $\dfrac{1628}{2}$

$\Rightarrow \dfrac{22}{7} \times \text{r} \times 37 = 814 \\[1em] \Rightarrow \text{r} = \dfrac{814 \times 7}{22 \times 37} \\[1em] \Rightarrow \text{r} = \dfrac{5698}{814} \\[1em] \Rightarrow \text{r} = 7 \text{ m.}$

⇒ r + h = 37

⇒ 7 + h = 37

⇒ h = 37 - 7

⇒ h = 30 m.

Volume of cylinder = πr²h

$= \dfrac{22}{7} \times 7^2 \times 30 \\[1em] = \dfrac{22}{7} \times 49 \times 30 \\[1em] = \dfrac{32340}{7} \\[1em] = 4620 \text{ m}^3.$

Hence, volume of solid cylinder is 4620 m³.