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Homework Help: Newton's Law of Cooling and ambient temperature

  1. Jul 8, 2010 #1
    I'm somewhat familiar with the formula:

    [tex]
    T(t)=T_{0}+(T_{i}-T_{0})e^{kt}
    [/tex]

    However, what if the ambient temperature is not constant? How would one find the temperature of an object with an ambient temperature that ramps from A to B (steady ramp let's say)?

    I'm honestly not sure how to solve for a certain time if the temperature is changing...

    Any advice would be greatly appreciated!
     
  2. jcsd
  3. Jul 8, 2010 #2

    kuruman

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    Science Advisor
    Homework Helper
    Gold Member

    To begin with, your exponential must be exp(-kt), otherwise it will blow up as time increases. I am sure this was a typo. Now for the big question. Where did the equation that you posted come from? Answer: It is the solution of the differential equation

    [tex]\frac{dT}{dt}=-k(T-T_0)[/tex]

    where T0 is the (constant) ambient temperature. If the ambient temperature is not constant but a function of time f(t), then you replace T0 with f(t) in the above differential equation and solve it (if you can).
     
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