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Math Help - find temp

  1. #1
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    Jan 2008
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    find temp

    A pan of water (46 degrees) was put into a fridge. Ten min. later, the temp was 39; 10 min. later, the temp was 33.
    Use newtons law of cooling to find the temp of the fridge..
    I know the equation.

    Do I need to find the rate, or how do I do this?
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  2. #2
    Junior Member
    Joined
    Nov 2009
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    By Newton's law of cooling you have that the temperature of the water is given by:
     T(t) = T_f + (46 - T_f) e^{-rt}
    where  T_f is the temperature of the fridge and r is some unknown rate constant. You are told T(10) = 39 and T(20) = 33, so:
     T(10) = 39 = T_f + (46 - T_f) e^{-10r}
     T(20) = 33 = T_f + (46 - T_f) e^{-20r}
    The first equation gives you:
     r = -1/10 ln(\frac{39-T_f}{46 - T_f})
    and the second gives you:
     r = -1/20 ln (\frac{33-T_f}{46 - T_f}) .
    Equating these two expressions for r and multiplying by -20 yields:
     2ln(\frac{39-T_f}{46 - T_f}) = ln (\frac{33-T_f}{46 - T_f}) .
    The 2 can move inside the log as a power, then you can get rid of the logs and solve. You should get  T_f = -3 .
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