grain is being poured through a chute at rate of 8 ft^3/min and forms a conical pile whose radius of the bottom is always 1/2 the height of pile. How fast is circumference increasing when the pile is 10 feet high?

Answer 1

given:
r = (1/2)h
2r = h

Let V be the volume

V = (1/3)π r^2 h
V = (1/3)π(r^2)(2r) = (2/3)π r^3
dV/dt = 2πr^2 dr/dt
when h = 10, r = 5
8 = 2π(25)dr/dt
dr/dt = 4/(25π)

Let circumference be C

C = 2πr
dC/dt = 2π dr/dt
so at that instance,
dC/dt = 2π(4/(25π) = 8/25 ft/min