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Law of cooling differential equation

  1. Feb 1, 2009 #1
    1. The problem statement, all variables and given/known data

    a) An object at 200 degrees F is put in a room at 60 degrees F.The temperature of the room decreases at the constant rate of 1 degree every 10 minutes. The body cools to 120 degrees F in 30 minutes. How long will it take for the body to cool to 90 degrees F?

    b) Show that the solution of the pertinent initial value problem which models the situation is:
    T(t) = 60 + 140e^(kt) + [(e^(kt) - kt - 1)/(10k)]

    c) Set-up an equation from which you can solve for k.

    d) Set-up an equation from which the required cooling time can be found.

    2. Relevant equations

    Newton's Law of Cooling: T'(t) = K(T(t) - T0)

    Note: T is in minutes

    3. The attempt at a solution

    a) This is variable seperable

    dT/dt = K(T(t) - T0)

    ∫dT/(T(t) - T0) = ∫k dt + C

    ln (T(t) - T0) = kt + C

    (T(t) - T0) = ce^(kt)

    T(t) = ce^(kt) + T0

    At T(0) = 200, and T0 = 60

    200 = ce^(K*0) + 60

    c = 140

    T(t) = 140e^(kt) + 60

    This is where I get stuck. I'm not really sure where to go next. I'm mainly confused by the fact that room temperature is decreasing as well.
  2. jcsd
  3. Feb 1, 2009 #2


    Staff: Mentor

    Yes, that's the problem. The differential equation you started with, dT/dt = K(T(t) - T0), assumes that the ambient temperature, T0, remains constant.

    The function that represents the ambient temperature is Ta = -t/10 + 60. You need to work that into the differential equation instead of T0.
  4. Feb 1, 2009 #3
    Thank you for your insight. I see where I need to go with this problem now.
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