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Thread: Rate of increase on conical pile

  1. #1
    Sep 2009

    Rate of increase on conical pile

    Sand is being dumped at a constant rate onto a conical pile, which is 5 feet high. When you see it again, 1 hour and 10 minutes later, the pile is 10 feet high. At what rate is the height increasing?

    If you can help me figure out this problem I'd appreciate it
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  2. #2
    Senior Member
    Dec 2008
    We may first write all of the information given in mathematical notation:

    $\displaystyle \begin{aligned}
    V&=\frac{1}{3}\pi r^2h\\

    In order that the problem have an answer, the ratio of the height to the radius, and thus the angle of the tip of the cone, must remain the same. (Otherwise, we could have the cone start out narrow and then get wider as $\displaystyle h$ approaches $\displaystyle 10$, lowering our answer for $\displaystyle \frac{dh}{dt}$ dramatically.) Therefore, we may say that

    $\displaystyle r=hk$

    for some $\displaystyle k$, and that

    $\displaystyle V=\frac{1}{3}\pi(hk)^2h=\frac{k^2\pi}{3}h^3.$

    In order to find the value of $\displaystyle \frac{dh}{dt}$ at $\displaystyle t=70$, in minutes, we may use the equations above together with the fact that

    $\displaystyle \frac{V(70)-V(0)}{70}=C.$
    Last edited by Scott H; Dec 1st 2009 at 03:20 PM.
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