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Math Help - DE: Temperature and Coffee

  1. #1
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    DE: Temperature and Coffee

    Problem:

    A freshly brewed cup of coffee has temperature 95C in a 20C room. When its temperature is 78 C, it is cooling at a rate of 1C per minute. When does this occur? Approximate to three decimal places.

    Here is my work, please verify if it is correct:


    T(t) = y0 * e^{kt} + Ts

    78 = 95*e^{kt}+20

    95e^{-t} = 78 - 20

     <br />
t = - ln(\frac {78 - 20} {95})<br />

    Therefore, the Coffee's temperature will be decreasing at 1 degree per minute after: 0.4934 minutes.
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  2. #2
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    Newton's Law of cooling ...

    \frac{dT}{dt} = k(T - T_a)

    \frac{dT}{dt} = k(T - 20)

    when T = 78 , \frac{dT}{dt} = -1

    k = -\frac{1}{58}


    \frac{dT}{T-20} = k \, dt

    \ln|T-20| = kt + C

    T-20 = Ae^{kt}

    T = 20 + Ae^{kt}

    at t = 0, T = 95 ...

    95 = 20 + A ... A = 75

    T = 20 + 75e^{kt}

    78 = 20 + 75e^{kt}

    58 = 75e^{kt}

    t = \frac{1}{k} \cdot \ln\left(\frac{58}{75}\right) \approx 15 minutes
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  3. #3
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    I see my mistake!

    Thanks a lot Skeeter,
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