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Thread: 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:


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

    $\displaystyle 78 = 95*e^{kt}+20$

    $\displaystyle 95e^{-t} = 78 - 20$

    $\displaystyle
    t = - ln(\frac {78 - 20} {95})
    $

    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 ...

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

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

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

    $\displaystyle k = -\frac{1}{58}$


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

    $\displaystyle \ln|T-20| = kt + C$

    $\displaystyle T-20 = Ae^{kt}$

    $\displaystyle T = 20 + Ae^{kt}$

    at $\displaystyle t = 0$, $\displaystyle T = 95$ ...

    $\displaystyle 95 = 20 + A$ ... $\displaystyle A = 75$

    $\displaystyle T = 20 + 75e^{kt}$

    $\displaystyle 78 = 20 + 75e^{kt}$

    $\displaystyle 58 = 75e^{kt}$

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

    Thanks a lot Skeeter,
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