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Differential Equation problem setup

  1. Oct 17, 2015 #1
    1. The problem statement, all variables and given/known data
    1. Water with a small salt content (5 lb in 1000 gal) is flowing into a very salty lake at the rate of 4 · 105 gal per hr. The salty water is flowing out at the rate of 105 gal per hr. If at some time (say t = 0) the volume of the lake is 109 gal, and its salt content is 107 lb, find the salt content at time t. Assume that the salt is mixed uniformly with the water in the lake at all times.

    2. Relevant equations
    In my setup I let w be the amount of water present in the lake (in gallons), t be time, and s be the amount of salt present in the lake (in pounds).

    for dw/dt I have 4x105 gal/hr coming in and 105 gal/hr leaving which gives me
    dw/dt = 3x105 gal/hr.

    my initial condition for w is w(0)= 109.

    my initial condition for s is s(0) = 107.

    getting an equation for ds/dt I have the amount of salt coming into the lake is (5/1000)(4x105) = 2x103 lb/hr. Let's call this equation (I).

    The amount of salt leaving the lake is given by:

    Currently I have (105)((107+2x103t)/109+3x105t) = (107+ 2x103t)/(104 + 3t). Let's call this equation (II).

    Therefore,
    ds/dt = (I)-(II).


    3. The attempt at a solution
    I think the main problem I am running into is my "amount of salt leaving the lake" equation. Something seems very sketchy about it, but I cannot think of any way to modify it. Also, it really bugs me that the problem statement says that "the salt content in the lake is always uniform". This makes me think I should always have a ratio of 10-2 lb/gal, which is giving me a lot of problems with my equation (II). I could easily solve the DE if I could just set it up.

    Any help would be greatly appreciated.
     
    Last edited: Oct 17, 2015
  2. jcsd
  3. Oct 17, 2015 #2

    SteamKing

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    Well, which statement is included in the actual problem statement?

    1. "Assume that the salt is mixed uniformly with the water in the lake at all times."
    or
    2. "the salt content in the lake is always uniform."

    These two statements don't necessarily mean the same thing.
     
  4. Oct 17, 2015 #3
    @SteamKing I suppose I may have been misinterpreting that statement. I read it as "Assume that the salt is mixed uniformly with the water in the lake at all times, which implies the salt content in the lake is always uniform." I think I see why A does not imply B in this case now, thank you for pointing that out.
     
  5. Oct 18, 2015 #4
    I actually figured it out with the help of my DE textbook, the upload_2015-10-18_12-14-7.png term should just be s, which actually makes a lot of sense because that is the part I was most unsure about.
     
    Last edited: Oct 18, 2015
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