let the height be h ft
then diameter = 3h ft, and the radius is 3h/2
V = (1/3)π r^2 h
= (1/3)π (9h^2/4)(h) = (3/4)π h^3
dV/dt = (9/4)π h^2 dh/dt
when h = 15, dV/dt = 10
10 = (9/4)π (15) dh/dt
dh/dt = 40/(135π) ft/s
= 8π/27 ft/s
At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the cone is approx imately three times the altitude. At what rate is the height of the pile changing when it is 15 feet high?
1 answer