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Math Help - Differential Equation

  1. #1
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    Differential Equation

    A tank contains kg of salt and L of water. A solution of a concentration kg of salt per liter enters a tank at the rate L/min. The solution is mixed and drains from the tank at the same rate.

    Find the amount of salt in the tank after hours.

    I got something like 36 kg and my equation was

    A= (320e^(-8t/1000)-320)/-8

    but i dont think this is correct
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  2. #2
    Senior Member vincisonfire's Avatar
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    Let C be the concentration of salt in the tank.
    Each minute, 0.04 kg/L * 8L = 0.32 kilos of salt goes in and C kg/L * 8 L = 8C goes out.
     \frac{dC}{dt} = 0.32 - C*8
     \frac{dC}{0.32 - C*8} = dt
     -ln(0.32 - C*8) = t + K
     ln(0.32 - C*8) = -t + K
     ln(0.32 - C*8) = -t + K
     0.32 - C*8 = e^{-t + K}
     C = 0.04 - Ke^{-t}
    Find K with initial condition C(0) = 0.08
     C = 0.04 + 0.04e^{-t}
    After 5 hours, C = 0.4 + almost nothing so Q = C*1000 = 40kg
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