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Homework Help: Differential Equations: Linear Models

  1. Feb 11, 2010 #1
    1. The problem statement, all variables and given/known data

    A 500-gallon tank originally contains 350 gallons of pure water. A 2-pound per gallon brine solution is pumped into the tank at 10 gallons per minute and the well-stirred mixture is pumped out at one-half that rate. How much salt is in the tank at th emoment of overflow?

    Answer: 510 lb

    2. Relevant equations

    dA/dt = [(rate in)(Concentration in)] - [(rate out)(concentration out)]

    3. The attempt at a solution

    Here is what I did.

    dA/dt = (2lb/gal * 10 gal/min) - [A/(350+5t) * 5]

    dA/dt = 20 - A/(70+t)

    Rearrange:

    dA/dt + A/(70+t) = 20

    Initial condition: A (0) = 0

    eintgral[1/(70+t)] = 70 + t

    A(t) = [1/(70+t)] * [10* integral(t+70)dt + c

    A(t) = 5(t+70) + c/(t+70)

    A(0) = 0;

    c = -24,500

    Thus; A(t) = 5(t+70) - 24500/(t+70)

    A (10)/500 = 400 - 24500/80

    = 93.75 lb/gal

    This answer is incorrect, as the answer should be 510 lb.

    PLEASE HELP!!

    Thanks!!
     
  2. jcsd
  3. Feb 11, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Why did you evaluate at t= 10? If there were initially 350 gallons in the 500 gallon tank, we need to add 150 gallons. With water coming in at a net 5 gallons per minute, that will take 150/5= 30 minutes, not 10. Also, of course, you give A in pounds per gallon. To find the number of pounds in the tank when it overflows, multiply that by 500 gallons.
     
  4. Feb 11, 2010 #3
    hi - thanks for help! :)

    But I am still confused;

    I now understand why t = 30 minutes. So, if I take t = 30, then I get:

    Thus; A(t) = 5(t+70) - 24500/(t+70)

    A (30)/500 = 500 - 24500/100

    = 500 - 245 = 255 lb/gal.

    But, if I multiply this 255 lb/gal by 500, I get ridiculously large number!! :) I get 130050 lb, which is incorrect..

    *I just observed that if I multiply 255 by 2, then I get the CORRECT answer: 510. But I get the wrong units. Also, it doesn't make sense to multiply 255 by 2 :)

    So, its the LAST part thats creating problem for me now!! :eek:
     
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