Question

At the instant when the radius of a cone is 3 inches, the volume of the cone is increasing at the rate of 9 pi cubic inches per minute. If the height is always 3 times the radius, find the rate of change of the radius at that instant.

Answer 1

dV/dt = 9 = surface area * dh/dt

9 = (pi r^2)(dh/dt)
but
h = 3 r
so
dh/dt = 3 dr/dt
so
9 = (pi r^2)(2 dr/dt)
but r = 3
so
9 = 9 pi (2 dr/dt
dr/dt = 1/2pi