The sum of the radius and the height of a cylinder is 37 cm and the total surface area of the cylinder is 1628 cm². Find the height and the volume of the cylinder.

Given, (h + r) = 37 cm, where h = height and r = radius of the cylinder.

Given, Total surface area = 1628 cm².

We know that Total surface area = 2πr(h + r).

∴ 2πr(h + r) = 1628

Putting values,

$\Rightarrow 2 \times \dfrac{22}{7} \times r \times 37 = 1628 \\[1em] \Rightarrow r = \dfrac{1628 \times 7}{2 \times 22 \times 37} \\[1em] \Rightarrow r = \dfrac{11396}{1628} \\[1em] \Rightarrow r = 7 \text{ cm}.$

Since, h + r = 37.

⇒ h + 7 = 37
⇒ h = 37 - 7 = 30 cm.

Volume of cylinder = πr²h.

Putting values we get,

$\text{Volume of cylinder } = \dfrac{22}{7}\times (7)^2 \times 30 \\[1em] = \dfrac{22 \times 49 \times 30}{7} \\[1em] = \dfrac{323400}{7} \\[1em] = 4620 \text{ cm}^3.$

Hence, the height of the cylinder = 30 cm and volume of cylinder = 4620 cm³