The ratio between the curved surface area and the total surface area of a right circular cylinder is 3 : 5. Find the ratio between the height and the radius of the cylinder.

It is given that the ratio between the curved surface area and the total surface area of a right circular cylinder is 3 : 5.

$⇒ \dfrac{\text {Curved surface area of cylinder}}{\text {Total surface area of cylinder}} = \dfrac{3}{5}\\[1em] ⇒ \dfrac{2πrh}{2πr(r + h)} = \dfrac{3}{5}\\[1em] ⇒ \dfrac{\cancel{2πr}h}{\cancel{2πr}(r + h)} = \dfrac{3}{5}\\[1em] ⇒ \dfrac{h}{r + h} = \dfrac{3}{5}\\[1em] ⇒ 5 \times h = 3 \times (r + h)\\[1em] ⇒ 5h = 3r + 3h\\[1em] ⇒ 5h - 3h = 3r\\[1em] ⇒ 2h = 3r\\[1em] ⇒ \dfrac{h}{r} = \dfrac{3}{2}\\[1em]$

Hence, the ratio between the height and the radius of the cylinder is 3 : 2.