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Math Help - Value of k (Newton's Law of Cooling)

  1. #1
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    Value of k (Newton's Law of Cooling)

    Hi
    I've been given the following question;
    Jane makes coffee with water from an urn which is at a temperature of 95 deg C.
    Jane carries a thermometer with her at all times and notes that 7 minutes later her coffee has cooled to 70 deg C.
    The room is not air-conditioned, and is 25 deg C.
    Suppose the temperature follows Newton’s Law of Cooling:

    where is the number of minutes after the coffee was made.
    It then asks me to find A, B and k. I know my answers for A and B are correct (25 and 70 respectively) but i am struggling to find k.
    It does then ask me to find the temperature after 10mins and the time for the coffee to reach 60degrees which i also have correct (62degrees and 11mins respectively). I used a value of k= ln(14/9) but it tells me this is incorrect.
    Am i doing something wrong?
    Thanks
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  2. #2
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    Re: Value of k (Newton's Law of Cooling)

    I'm not sure how you got B=70 but that is not correct.
    You have 3 equations to solve for the 3 unknowns
    1. Jane makes coffee with water from an urn which is at a temperature of 95 deg C.
    At time t=0 the temperature is 95

    2. Jane carries a thermometer with her at all times and notes that 7 minutes later her coffee has cooled to 70 deg C.
    At time t=7 the temperature is 70

    3. As time time tends to infinity the temperature of the coffee becomes the same as the room temperature.
    when t=\infty, T=25

    The equation for the third one is
    25=A+Bk^{\infty}

    There are three cases for k,
    k>1 this would cause the temperature to grow to infinity as time passes which is clearly wrong
    k=1 this would mean that T(t)=A+B which is constant, clearly temperature is not constant so k cannot be 1
    k<1 is the only reasonable case left

    If k<1 then k^{\infty}=0 so 25=A+Bk^{\infty} simplifies to 25=A

    Now you have A, use the other two equations to find B and k. Once you have them the second part of your question should follow easily.
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