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Math Help - Mixture problem

  1. #1
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    Mixture problem

    A tank initially contains 60 gallons of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2 gal/min, and the perfectly mixed solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour.

    I am asked to
    A) Find the amount of salt in the tank after t minutes
    B) What is the maximum amount of salt ever in the tank

    I am confused as to how to set this up and then after I set it up how would I determin B?

    Thanks
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    A tank initially contains 60 gallons of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2 gal/min, and the perfectly mixed solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour.

    I am asked to
    A) Find the amount of salt in the tank after t minutes
    B) What is the maximum amount of salt ever in the tank

    I am confused as to how to set this up and then after I set it up how would I determin B?

    Thanks
    Here (a). Let A be the amount of salt at time t. So

    \frac{dA}{dt} = r_i c_i - r_o c_o

    rates in and out and concentrations in and out. For your problem since the tank is emptying, V, the colume is changing so

    V = V_0 + (r_i - r_o) t = 60 - t

    \frac{dA}{dt} = 2 \cdot 1 - 3 \cdot \frac{A}{60-t},\;\;\;A(0) = 0 - a ODE to solve.
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