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Math Help - [SOLVED] Exponential models

  1. #1
    friend37866
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    [SOLVED] Exponential models

    The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by friend37866 View Post
    The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.
    This is related to Newton's Law of Cooling, I presume. (See part way down the page of this link.)

    We have that
    T(t) = T_{env} + (T(0) - T_{env})e^{-t/t_0}
    where T(0) is the temperature of the body (in this case the thermometer), T_{env} is the temperature of the environment, and t_0 is the "time constant," which we don't know yet.

    We know that T(0) = 21~^oC and that at t = 5~min, T(5) = 16~^oC, and at t = 10~min, T(10) = 13~^oC.

    So:
    T(5) = 16 = T_{env} + (21 - T_{env})e^{-5/t_0}
    and
    T(10) = 13 = T_{env} + (21 - T_{env})e^{-10/t_0}

    Two equations and two unknowns. We want T_{env} and don't care about the time constant, so what I recommend you do is, instead of trying to solve for t_0 directly, solve the top equation for e^{-5/t_0}. Then note that e^{-10/t_0} = \left ( e^{-5/t_0} \right )^2 and use that for your substitution. You'll be left with an equation for T_{env}.

    Sorry, I gotta run.

    -Dan
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