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, its volume is :

1. 3180 m³

2. 4620 m³

3. 5240 m³

4. None of these

Let radius be r m and height be h m.

Given,

r + h = 37 m …(1)

Totals surface area of cylinder = 1628 m²

⇒ 2πr(r + h) = 1628

⇒ 2πr × 37 = 1628

$\Rightarrow 74 \times \dfrac{22}{7} \times \text{r} = 1628 \\[1em] \Rightarrow \dfrac{1628}{7} \times \text{r} = 1628 \\[1em] \Rightarrow \text{r} = \dfrac{1628 \times 7}{1628} \\[1em] \Rightarrow \text{r} = 7 \text{ m.}$

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

⇒ r + h = 37

⇒ 7 + h = 37

⇒ h = 37 - 7

⇒ h = 30 m.

Volume of cylinder = πr²h

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

Hence, option 2 is the correct option.