Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature an

Question

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 190190 degrees Fahrenheit when freshly poured, and 11 minutes later has cooled to 172172 degrees in a room at 6060 degrees, determine when the coffee reaches a temperature of 122122 degrees.

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Linh Đan 4 years 2021-09-04T23:31:56+00:00 1 Answers 7 views 0

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  1. kimchi2
    0
    2021-09-04T23:33:11+00:00
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    Answer:

    4.9 minutes

    Explanation:

    Given; T(t) = Ce^-kt + Ts

    Now;

    T(t) = 190 degrees Fahrenheit

    Ts = 60 degrees

    To obtain C;

    190 = Ce^0 + 60

    190 – 60 = C

    C = 130

    Hence, to find k when t=11

    172 = 130 e^-11k + 60

    172 -60/130 = e^-k

    e^-k = 0.86

    ln(e^-k) = ln( 0.86)

    -k = -0.15

    k = 0.15

    Hence at 122 degrees, t is;

    T(t) = Ce^-kt + Ts

    122 = 130e^-0.15t + 60

    122 – 60/130 = e^-0.15t

    0.477 = e^-0.15t

    ln (e^-0.15t) = ln (0.477)

    -0.15t = -0.74

    t = 0.74/0.15

    t = 4.9 minutes

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