As sand leaks out of a hole in a container it forms a conica

As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as the radius. If the height of the pile is increasing at a rate of 6 inches/minute, find the rate at which the sand is leaking out when the altitude is 10 inches.

Solution

V= volume of cone r= radius of cone h= height of cone Rates: dh/dt = 6 in/min dr/dt = (is this also 6 in/min since the radius is always the same as the height?) dV/dt = unknown V= (1/3)pi*r^2*h r=h, so V= (1/3)pi*(h^2)*h = (1/3)pi*(h^3) dV/dt = pi(h^2)(dh/dt) h=10, dh/dt= 6 dV/dt = pi(100)(6) = 600 pi or 1884.954 in/min