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Math Help - Rate of change problem - please help

  1. #1
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    Rate of change problem - please help

    1. A dog eats her food out of a square based rectangular prism with both lengths and width of 15cm and a height of 10cm. As soon as the box is totally empty, a machine automatically refills it at a rate of 3cm^3/s. Find the rate of change of the height of the food within the box 2 seconds after it begins filling.

    Okay, I know how to do most of this question, the only thing im confused about is what to do with the time = 2s... Any help would be great, thanks.
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    Quote Originally Posted by sugar_babee View Post
    1. A dog eats her food out of a square based rectangular prism with both lengths and width of 15cm and a height of 10cm. As soon as the box is totally empty, a machine automatically refills it at a rate of 3cm^3/s. Find the rate of change of the height of the food within the box 2 seconds after it begins filling.

    Okay, I know how to do most of this question, the only thing im confused about is what to do with the time = 2s... Any help would be great, thanks.
    We have a base area of 15~\text{cm} \times 10~\text{cm} = 150~\text{cm}^2

    So the volume of food in the bowl at any time after the machine starts filling it is
    V = 150h

    Thus
    \frac{dV}{dt} = 150 \frac{dh}{dt}

    We know the food is coming in at 3~cm^3/s, so that's dV/dt.

    Thus
    \frac{dh}{dt} = \frac{\frac{dV}{dt}}{150~cm^2} = \frac{3~cm^3/s}{150~cm^2} = \frac{1}{50}~cm/s

    Since the rate of height change is linear with respect to the rate of volume change, it happens (in this particular case) to be independent of the time.

    -Dan
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    so the time they give doesnt have anything to do with the answer? And wouldnt the base be 15 x 15 =225cm? since the height is 10 not the width?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sugar_babee View Post
    so the time they give doesnt have anything to do with the answer?
    Correct.

    You would need the time if your function for \frac{dh}{dt} depended on time. Since it doesn't, you don't need the time.

    Quote Originally Posted by sugar_babee View Post
    And wouldnt the base be 15 x 15 =225cm? since the height is 10 not the width?
    Whoops! Sorry about that. You are correct. The volume formula should have been
    V = 225h

    My apologies.

    -Dan
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