utf29-001

{Newton's Law of Cooling} states that the rate of change of the temperature T of an object is proportional to the difference between T and the temperature $\tau$ of the surrounding medium: $$\frac{d T}{dt} = k (T-\tau)$$ where k is a constant.

1. Solve this equation for T. (the general solution)
2. Solve this equation for T given that $ T(0) = T_0.$
3. Take $T_0 > \tau.$ What is the limiting temperature to which the object cools as t increases? What happens if $T_0 < \tau$?

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Tags: differential-equations, limits, limits-at-infinity, exp, physics-application, multistep, a--a, t2, c2

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See the discussion on Newton’s Law of Cooling!

– Jeff Hildebrand 1998-08-09

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Students may have trouble figuring out the sign of k, but it’s very important that they make the connection between “cooling” and a negative derivative.

– Concha Gomez 1999-02-19