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Homework Help: Differential Equations: word problem

  1. Jan 23, 2010 #1
    1. The problem statement, all variables and given/known data

    When a cake is removed from an oven, its temperature is measured at 300 degrees F, 3 min later its 200, how long will it take for it to cool off to 75? room temp is 70. Assume newtons law of cooling applies. The rate of cooling is proportional to the difference between the current temp and the ambient temp.

    2. Relevant equations

    This is what i did:
    dx/dt = k(x-70)

    3. The attempt at a solution

    dx/dt = k(x-70)
    integral 1/(x-70) dx = integral k dt
    ln|x-70| = kt+c
    after some algebra....
    x = C*e^(kt) +70 where C = + or - e^c
    at x(o) = 300 so...
    300 = C*e^(k*0) +70
    which gives me C = 230

    x(t) = 230*e^(kt) +70
    x(3) = 200
    200 =230*e^(k*3) +70
    k = 1/3 * ln(13/23)

    now that i have an equation i can solve for the time it takes to get to 75
    i'll leave k as k since its confusing when put into the equation
    75 = 230*e^(kt) +70
    i solved and got
    t = (3*ln(5/230))/(ln(13/23))
    which is approximately 20 minutes

    Did i do this right? or did i go wrong somewhere, i just want to make sure im getting the right idea with these kinds of problems
  2. jcsd
  3. Jan 24, 2010 #2
    mathematically i find no mistake

    but physically speaking we are formulating an equation for cooling so it is better to write the equation as
    dx/dt = -k(x-70)
    so that you will get a positive k value in the end of the calculation

    because if k value is given as positive in the problem
    your steady state result would tend to infinity instead of room temp 70.
    x = C*e^(kt) +70
  4. Jan 24, 2010 #3


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    Science Advisor

    No, it does not- his k is (1/3)ln(13/23), a negative number.
  5. Jan 24, 2010 #4
    what i wanted to convey is these solutions have a steady state and transient state.

    the transient state should tend to zero when time approaches infinty.(for this cooling problem)

    x(t) = 230*e^(kt) +70, k<0 should be the solution if you have right idea about this problem
  6. Jan 24, 2010 #5
    Thank you so much. I didn't think it would make a difference if it was -k or not because i saw an example somewhat similar to this in the book i'm using and it had a positive k. The answer just looked very confusing so i wanted to make sure i doing the question right
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