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Newton's Law of cooling DE involving delicious coffee.

  1. Feb 4, 2012 #1
    I did this problem a few times now and double checked my procedure but I don't see what I did wrong. The answer should be 6.07 minutes according to the back of the book, but I get 3.63.

    1. The problem statement, all variables and given/known data
    Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 200 degrees F when poured, 1 minute later cooled to 190 degrees F, room temperature 70 deg F, determine time at which temperature is 150 degrees F.


    2. Relevant equations
    dT/dt = k(Ts - T(t))

    k being constant of proportionality.
    Ts ambient temperature
    T(t) temperature of cup of coffee at a time t.

    T(0) = 200
    T(1) = 190


    3. The attempt at a solution

    dT/dt + kT(t) = kTs

    multiplying equation by integration factor mu = e^(kt)

    (T(t) * e^(kt))' = k*Ts*e^(kt)

    Integrating both sides:

    T(t) * e^(kt) = Ts*e^(kt) + c
    T(t) = Ts + c*e^(-kt)

    Since Ts = 70, and T(0) = 200:

    200 = 70 + c
    c = 140

    T(t) = 70 + 140e^(-kt)

    T(1) = 190:

    190-70 = 140 e^(-k)
    6/7 = e^(-k)

    ln(7/6) = k

    Therefore, equation for temperature of coffee at any time t:
    T(t) = 70 + 140e^(ln(6/7)t)

    Solving for T(t) = 150:

    (150-70)/140 = e^(tln6/7)
    ln(4/7) = tln(6/7)
    t = ln(4/7)/ln(6/7) = 3.63 minutes.


    Can anyone see if maybe I made a mistake? Thanks a ton.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 4, 2012 #2

    ehild

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    200-70=????

    ehild
     
  4. Feb 4, 2012 #3
    Hehehe. Looks like I need to go back to arithmetic class. XD Such a insignificant part of the calculation it seemed that I overlooked it several times. Thanks a ton.
     
  5. Feb 4, 2012 #4

    ehild

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    Homework Helper
    Gold Member

    Such things happen to everybody including myself. :biggrin:

    ehild
     
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