Calculus Related Rates Conical Pile of Sand Problem

Given that h=r/2, the volume formula is

V = pi r^3 / 6
dV/dt = dV/dr * dr/dt
dV/dr = pi r^2 / 2
When h = 40in, r = 80in so dV/dr = 3200 pi in^2
r = 2h so dr/dt = 2 dh/dt = 6 in/min
dV/dt = (3200 pi in^2)(6 in/min) = 19200 in^3/min

So your answer is correct but your units are slightly off.

Here's the way I'd do the problem

V=1/3*pi*r^2*h

h=r/2 or r=2h

So

V=1/3*pi*(2h)^2*h=4/3*pi*h^3

differentiate

dV/dt=4/3*pi*3*h^2*dh/dt

dV/dt=4*pi*h^2*dh/dt

putting in values h=40 in & dh/dt=3 in/m

Then dV/dt=4*3.14159*(40 in)^2*3 in/m ~ 60318.5 in^3/s = 1.29 yd^3/m

Rap