Sand is falling off a conveyor belt onto a conical pile at a

Sand is falling off a conveyor belt onto a conical pile at a rate of 15 cubic feet per minut.

The diameter ofthe base of the cone is twice the altitude. At what rate is the hight of the pile changing when it is 10 feet high?

Solution

Volume of cone = (1/3) *r²h
Given d = 2h; i.e 2r = 2h; => r = h
Hence volume v = (1/3)h³
Now dv/dt = (1/3)*3h² * dh/dt = h²dh/dt
Now dv/dt = 15 cubic feet/min; And at that instant h = 10 feet

Hence 15 = *10²*dh/dt
Hence dh/dt = 15/(100) = 0.0477 feet/min