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Journal Article

Cooling Coffee without Solving Differential Equations

Robert Israel, Peter Saltzman and Stan Wagon
Mathematics Magazine
Vol. 86, No. 3 (June 2013), pp. 204-210
DOI: 10.4169/math.mag.86.3.204
Stable URL:
Page Count: 7
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Cooling Coffee without Solving Differential Equations
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Summary A classic textbook problem is to show, assuming Newton’s law of cooling, that if cold milk is added to coffee that has been cooling down, the result will be colder than if the milk was added at an earlier time. We formulate and prove a theorem that shows this holds when the linear function of Newton’s law is replaced by any function satisfying a certain weak convexity condition. This is relevant to the real-world problem, since Newton’s law is not an adequate model for cooling liquids; it ignores the large amount of heat loss due to evaporation, as well as the smaller loss due to radiation.

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