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1MileCrash
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Homework Statement
Sand is pouring from a pipe at the rate of 16 cubic feet per second. If the falling sand forms a conical pile on the ground whose altitude is always 1/4 the diameter of the base, how fast is the altitude increasing when the pile is 4 feet high?
Homework Equations
[itex]V = \frac{1}{3}\pi r^{2}h[/itex]
The Attempt at a Solution
Firstly, since I am given that the altitude is 1/4 of the diameter, I rearrange the volume formula.
1/4(2)r = h
1/2r = h
r = 2h
So:
[itex]V = \frac{1}{3}\pi 2h^{2}h[/itex]
or
[itex]V = \frac{1}{3}\pi 2h^{3}[/itex]
Second, I'm given the rate of volume increase as 16, so [itex]\frac{dV}{dt}=16[/itex]
Differentiation of both sides of the volume equation gives me:
[itex]16 = 2\pi 2h^{2}\frac{dh}{dt}[/itex]
So it follows that
[itex]\frac{dh}{dt}= \frac{16}{2\pi h^{2}}[/itex]
Solving this for h = 4 gives me ~= .159155
Which is incorrect.
Can someone show me where I'm messing this up?
Thanks!