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Brine-ish Question with Linear Equations

  1. Mar 26, 2007 #1
    1. The problem statement, all variables and given/known data

    we have a tank with 400L of water with Cl, 0.02kg of Cl. Fresh water is pumped in at 4L/s and out at 10L/s. Find the amount of Cl in the tank as a function of t.

    2. Relevant equations

    Use of differential equations

    3. The attempt at a solution

    I set up V(t)=400-6t in Liters and seconds. then I want
    dy/dx=(rate in)-(rate out) for Cl flowing. But no Cl is coming in correct? So rate in ends up being zero, which confuses me because then everything ends up equalling zero from what I've done.

    If you want to see more of what I've done, please post. Am I wrong to say rate in is 0???
  2. jcsd
  3. Mar 26, 2007 #2
    rate in= (4L/s)(0g/L)=0
    rate out=(10L/s)(y(t)/(1000-6L))
    then dy/dt-(10y)/(1000-6t)=0
    I(t)=(1000-6t)^-10/6 (took a few steps to find, but I think it's right).
  4. Mar 26, 2007 #3
    sorry...it seems like a lot of people have looked at my problem...what is unclear about it? Why can't anyone help? Is there something I could do?
  5. Mar 26, 2007 #4
    I end up with integral ((400-6t)(y))prime=integral 0dt, and that can't be right.
  6. Mar 27, 2007 #5


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    Staff Emeritus
    Science Advisor

    Why would "everything end up being 0"? Yes, there is no Cl flowing in, but there certainly is Cl flowing out. Let X(t) be the amount of Cl in the tank at time t seconds. The volume of water at that time is, as you say, 400-6t so the concentration of Cl is X/(400-6t) kg/liter. Since the water is flowing out at 10 l/min, Cl is flowing out at 10X/(400-6t) kg/min.
    dX/dt= -10X/(400-6t). Looks like a simple separable equation to me.

    By the way, don't get upset if no one responds within a few minutes- some of us have lives to live!
  7. Mar 27, 2007 #6
    Thanks! I wondered if it was a separable equation, but because the rate in is different from the rate out, isn't it linear, but not separable? That was sort of also holding me back...
  8. Mar 27, 2007 #7


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    Staff Emeritus
    Science Advisor

    Actually, it is both linear and separable.
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