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Related rates grain hopper problem

  1. Nov 12, 2004 #1
    Grain is emptying from the bottom of a funnel-shaped hopper at a rate of 1.2m3/min. If the diameter of the top of the hopper is 5m and the sides make an angle of 30o with the vertical, determine the rate at which the level of grain in the hopper is changing, when it is half the height of the hopper.
    P.S. (I am supposed to use implicit differentiation to find rates of change)

    The answer is -0.24m/min.

    I can't get it!!
  2. jcsd
  3. Nov 13, 2004 #2


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    You can use explicit differentiation.

    The angle is 30 degrees. The tangent of 30 degrees equals the radius divided by the heighth. In other words:

    [tex]h=\frac{r}{tan 30}[/tex]

    The angle stays constant, so tan 30 is also a constant.

    As the grain empties, it's cross section is always a similar triangle (similar to the shape of the hopper). If the heighth of the hopper is one-half, so are all the other sides.

    Edit: I thought you were asking if you had to use implicit differentiation.

    The 1/3 pi is just a constant. You apply the product rule to differentiate hr^2 (keeping in mind you have to apply the chain rule, as well).

    To get an answer, you still have to eventually find h from r.
    Last edited: Nov 13, 2004
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