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# Calculus Related Rates Conical Pile of Sand Problem

Sat 19 May, 2012 06:20 pm
Sand poured on the ground forms a conical pile whose altitude is always one half the radius of the base. When the height of the pile is 40 in. it's increasing at 3 in/min. How fast is the sand falling?
(Use v=1/3pi(r^2)h
I got 19,200pi in/min as my answer. I'm not sure it's correct though.
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engineer

1
Sat 19 May, 2012 07:00 pm
@suemonsta101,
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

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raprap

1
Sat 19 May, 2012 07:18 pm
@suemonsta101,
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

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