Question

The radius of a right circular cylinder is increasing at the rate of 5cm/s but its total surface area remains constant at 800cm². At what rate is the height changing when the radius is 10cm? Please enter your answer in decimal format with three significant digits after the decimal point.

Answer 1

A = 2 π R^2 + 2 π R H total area of cylinder

A / (2 π R) = R + H at given conditions

H = 800 / (2 π 10) - 10 = 800 / 62.832 - 10 = 2.732

dA / dt = 2 π (2 R dR/dt+ R dH/dt + H dR/dt) from first equation

0 = 2 R dR/dt + R dH/dt + H dR/dt

R dH/dt = -(2 R dR/dt + H dR/dt)

dH / dt = -dR/dt (2 + H / R)

dH / dt = -5 (2 + 2.732 / 10) = -5 * (2 + .2732) = -11.366