Question

sand is being dragged at the rate of 10m^3/min into a conical pile. if the height of the pile is always twice the base radius, at what rate is the height increasing when the pile is 2m high?

Answer 1

The volume of anything pointy with straight sides is (1/3) base area * height
so
V = (1/3) pi r^2 h
but h = 2 r
so
V = (2/3) pi r^3
find dr/dV, worry about dh/dV later (note h = 2 r)

dV = (2/3) pi (3 r^2) dr = 2 pi r^2 dr
dV/dt = 2 pi r^2 dr/dt = 10 m^3/min
but r = h/2 and dr/dt = (1/2)dh/dt
so
10 = 2 pi h^2/4 (1/2) dh/dt

40 = pi h^2 dh/dt
when h = 2
40 = 4 pi dh/dt
so
dh/dt = 10/pi meters/minute