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First-Order Linear Differential Equation

  1. Jan 28, 2014 #1
    1. The problem statement, all variables and given/known data
    A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing 5 parts per billion flows into the top of the tank at a rate of 4 gallons per minute. The water in the tank is kept well mixed, and 2 gallons per minute are removed from the bottom of the tank. How much dioxin is in the tank when the tank is full?


    2. Relevant equations

    I'm going to use D(t) as the amount of dioxin.

    3. The attempt at a solution

    dD/dt = (5)(4) - 2*(D(t)/(200+2t))

    Using an integrating factor of t + 100, and the initial condition of D(0) = 2, I got that D(t) = (10t2 + 2000t + 200)/(t+100)

    But when using t = 100, I get the answer to be 1501 ppb instead of the more appropriate 4.25 ppb. Where did I go wrong?
     
  2. jcsd
  3. Jan 28, 2014 #2

    vela

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    You need to be careful with the units. Use D(t) to denote the number of gallons of dioxin in the tank rather than the amount of dioxin in ppb.
     
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