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Math Help - Mixture problem

  1. #1
    MHF Contributor alexmahone's Avatar
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    Mixture problem

    A tank initially contains 60 gal 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 h.

    (a) Find the amount of salt in the tank after t minutes.
    (b) What is the maximum amount of salt ever in the tank?

    My working:

    Let x(t) (in lb) be the amount of salt in the tank at time t.

    (a) \frac{dx}{dt}=r_ic_i-r_oc_o=2*1-3*\frac{x}{60-t}=2-\frac{3x}{60-t}

    \frac{dx}{dt}+\frac{3x}{60-t}=2

    \rho(t)=exp\left(\int \frac{3}{60-t}dt\right)=exp(-3ln\ (60-t))=\frac{1}{(60-t)^3}

    Multiplying by \rho(t), we get

    \frac{1}{(60-t)^3}\frac{dx}{dt}-\frac{3x}{(60-t)^4}=\frac{2}{(60-t)^3}

    \frac{d}{dt}\left[\frac{x}{(60-t)^3}\right]=\frac{2}{(60-t)^3}

    \frac{x}{(60-t)^3}=2\frac{(60-t)^{-2}}{2}+C

    \frac{x}{(60-t)^3}=\frac{1}{(60-t)^2}+C

    x=(60-t)+C(60-t)^3

    x(0)=0

    0=60+C*60^3

    C=-\frac{1}{3600}

    x=(60-t)-\frac{(60-t)^3}{3600}

    Solved it! Don't bother.
    Last edited by alexmahone; September 10th 2011 at 05:32 AM.
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  2. #2
    MHF Contributor

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    Re: Mixture problem

    Aww, you're just showin' off!
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