Gravel is being dumped from a conveyor belt at a rate of 10 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 7 ft high? (Round your answer to two decimal places.)
let the radius of your cone be r ft
since the height is the same, the height is r ft
V = (1/3)π r^2 h, in this case h = r , so clearly dh/dt = dr/dt
V = (1/3)π r^3
dV/dt = π r^2 dr/dt
when h = 7, r = 7, and
10 = π (49) dr/dt , (skipping units for convenience sake)
dr/dt = 10/(49π) , but I said dh/dt = dr/dt, so
dh/dt = 10/(49π) ft/s or appr .06 ft/s