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Math Help - Can't figure out how to set up this modeling problem

  1. #1
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    Can't figure out how to set up this modeling problem

    So I feel confident that I can solve the differential equation once I set it up, however I'm not able to set up the problem.

    The question is as follows:

    Pure mountain water flows into a 10 million gallon lake at a reate of 3 million gallons
    per day. Water flows out of the lake at the same rate. Suppose that someone dumps 1000 pounds of toxic substance YUK into the lake. Assume that YUK mixes evenly with the lake water instantly.

    a). Write down and then solve the initial value problem for Y (t), the total amount of YUK in the lake as a function of the time t in days after the dumping occurred.


    For these types of problems I usually set it up as a function Q' = rate in - rate out
    so for this problem I tried setting it up as Q' = -3,000,000 (concentration of water leaving lake) as there is no rate of YUK in. I can't figure out the concentration of the water flowing out however

    Any help is appreciated

    EDIT: On further thinking I decided to put up what I thought might work, if this is right please let me know and I'll take this question down:

    Y'(t) = -3,000,000Y(t) / 10,000

    I realized the rate out is -3,000,000 gallons, and I assumed the concentration of Yuk in the lake is 1/10,0000(Y(t)) lb / gal which would come out
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  2. #2
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    Quote Originally Posted by addieandjack View Post
    So I feel confident that I can solve the differential equation once I set it up, however I'm not able to set up the problem.

    The question is as follows:

    Pure mountain water flows into a 10 million gallon lake at a reate of 3 million gallons
    per day. Water flows out of the lake at the same rate. Suppose that someone dumps 1000 pounds of toxic substance YUK into the lake. Assume that YUK mixes evenly with the lake water instantly.

    a). Write down and then solve the initial value problem for Y (t), the total amount of YUK in the lake as a function of the time t in days after the dumping occurred.


    For these types of problems I usually set it up as a function Q' = rate in - rate out
    so for this problem I tried setting it up as Q' = -3,000,000 (concentration of water leaving lake) as there is no rate of YUK in. I can't figure out the concentration of the water flowing out however

    Any help is appreciated

    EDIT: On further thinking I decided to put up what I thought might work, if this is right please let me know and I'll take this question down:

    Y'(t) = -3,000,000Y(t) / 10,000

    I realized the rate out is -3,000,000 gallons, and I assumed the concentration of Yuk in the lake is 1/10,0000(Y(t)) lb / gal which would come out
    \displaystyle\frac{dA}{dt}=R_{\text{in}}-R_{\text{out}}

    All my numbers are scaled in millions.

    R_{\text{in}}=3\text{gals/day}\cdot 0\text{Yuk}=0

    \displaystyle R_{\text{out}}=3\text{gals/day}\cdot \frac{.001}{10}\text{Yuk Lbs/gal}=\frac{.003}{10}

    \displaystyle\frac{dA}{dt}=-\frac{\frac{.003}{10}A}{10}\Rightarrow A'+.00003A=0

    P(t)=.00003, \ \ Q(t)=0

    \displaystyle t\exp{\left(\int P(t)dt\right)}=\int\left[\exp{\left(\int P(t)dt\right)}Q(t)\right]dt
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