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Math Help - Rates

  1. #1
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    Rates

    A conical paper cup 4inches across the top and 6 inches deep leaks water at a rate of 1 cubic inch per minute. at what rate does the water level drop (a) when the water is 3 inches deep? (b) when the cup is half full?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by mathletsgo View Post
    A conical paper cup 4inches across the top and 6 inches deep leaks water at a rate of 1 cubic inch per minute. at what rate does the water level drop (a) when the water is 3 inches deep? (b) when the cup is half full?
    a) It's given that \frac{dV}{dt}=-1. We know that V = \frac{2\pi rh}{3} and \frac{dV}{dh} = \frac{2\pi r}{3}. We also know that, by similar triangles, h=3 implies that r=1.

    Putting it all together, we have:
    \frac{dV}{dt}=\frac{dV}{dh}\cdot\frac{dh}{dt} \implies -1 = \frac{2\pi\cdot 1}{3}\cdot\frac{dh}{dt} \implies \frac{dh}{dt}=-\frac{3}{2\pi}
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