The radius of a right circular cylinder is increasing at the rate of 5 cm/s

Question

The radius of a right circular cylinder is increasing at the rate of  but its total surface area remains constant at  At what rate is the height changing when the radius is ?

Answer 1

dr/dt = 5 cm/s

surface area = 2πrh + 2πr² = 2πr(h + r) = 800 cm²

h = 400/πr – r

dh/dt = –400/πr²dr/dt – dr/dt = –(400/πr² + 1)dr/dt

Plugging in r = 10, dr/dt = 5

dh/dt = –(400/π(10)² + 1)(5) = –(4/π + 1)(5)

≈ –11.366 cm/s

Answer (arbitrarily rounded)

When the radius is 10 cm, the height is changing at a rate of –11.4 cm/s