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:

The sum of the radius and the height of a cylinder is 37 cm.

Total surface area of the cylinder = 1628 cm²

Let r be the radius and h be the height of cylinder.

So, r + h = 37

As we know that total surface area of cylinder = 2πr(r + h)

$⇒ 2 \times \dfrac{22}{7} \times r \times 37 = 1628\\[1em] ⇒ \dfrac{1628}{7} \times r = 1628\\[1em] ⇒ r = \dfrac{7 \times 1628}{1628}\\[1em] ⇒ r = \dfrac{7 \times \cancel{1628}}{\cancel{1628}}\\[1em] ⇒ r = 7$

It is given that r + h = 37

⇒ 7 + h = 37

⇒ h = 37 - 7

⇒ h = 30

Height of the cylinder is 30 cm.

Volume of the cylinder = πr²h

$= \dfrac{22}{7} \times 7^2 \times 30\\[1em] = \dfrac{22}{7} \times 49 \times 30\\[1em] = \dfrac{32,340}{7}\\[1em] = 4,620$

Hence, the height of the cylinder is 30 cm and the volume is 4,620 cm³.