In a right circular cylinder, the ratio of the curved surface to the total surface area is 3:7. Find the ratio of the height of the cylinder to the radius of its base.

3 : 4

Curved surface of cylinder = 2\(\pi \)rh

Total surface area of cylinder = (2\(\pi \)rh + 2\(\pi \)r² 

According to the question,

⇒ \(\frac{CSA}{TSA}\) = \(\frac{3}{7}\)

⇒ \(\frac{h}{h + r}\) = \(\frac{3}{7}\)

⇒ 7h = 3h + 3r

⇒ 7h - 4h = 3r

⇒ 4h = 3r

⇒ \(\frac{h}{r}\) = \(\frac{3}{4}\)

⇒ h : r = 3 : 4

Therefore, the ratio of height and radius is 3 : 4.