Two coffee drinkers pour themselves a cup of coffee each just after the kettle has boiled. The woman adds milk from the fridge, stirs it in and then waits for it to cool. The man waits for the coffee to cool first, then just before drinking adds the milk and stirs. If they both begin drinking at the same time, whose coffee is cooler? Justify this answer mathematically.
Assume that the air temperature is colder than the coffee and the milk is even colder. Also assume that after the milk is added and stirred, the temperature drops by a fixed percentage.
My attempt: The way I think about it is, the man that pours the milk will have the cooler coffee. This is because the woman's coffee will cause the cold milk to heat up. In contrast, if the man lets the coffee heat to dissipate first, then the heat will be reduced, so when the cold milk is poured in, it will provide a burst of coolness, and the cold milk will also not be heated to such a high temperature. I believe that this could have something to do with the Laws of Thermodynamics. I wonder whether someone has a mathematical explanation for this problem.
Suppose the first person adds enough milk so the temperature is exactly room temperature. Then clearly if the second person adds the same amount his coffee will be colder. Now if things are continuous (and they should be!) the result should still hold if slightly less milk is used. Now I have no idea if things at some point would change but the problem seems to indicate that is not the case.
Edit: More realistic situation with the same result, the man adds enough milk to have exactly room temperature coffee. The woman's coffee will not reach room temperature in finite time assuming a purely mathematical model ;)
Edit2: Anyone want to explain the down votes?