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Derivatives, rates of change (cone)

  1. Dec 2, 2013 #1
    1. Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

    2. Relevant equations
    $$V=\frac{\pi}{3}r^2h$$
    3. The attempt at a solution

    Diameter = height, so $$\frac{h}{2}=r$$
    $$V=\frac{\pi}{3}\frac{h^2}{4}h = \frac{\pi}{12}h^3$$
    $$\frac{dV}{dt}=\frac{\pi}{12}(3×h)\frac{dh}{dt}$$
    $$30=\frac{\pi}{12}(3×10)\frac{dh}{dt}$$
    $$\frac{dh}{dt}=\frac{12}{\pi}$$

    The textbook's answer is $$\frac{6}{5\pi}$$ What did I do wrong?
     
    Last edited: Dec 2, 2013
  2. jcsd
  3. Dec 2, 2013 #2

    SteamKing

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    You incorrectly assumed that the diameter of the pile = half of the radius.
     
  4. Dec 2, 2013 #3
    Sorry that was just a typo. The work should still follow r=h/2.
     
  5. Dec 2, 2013 #4

    SteamKing

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    Check your differentiation.
     
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