seanieg89
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A well known but nice practical question:
You have a boiling cup of coffee (so 100 degrees) and you wish to drink it in exactly 5 minutes time. You like to dilute your coffee with milk (at say 5 degrees) in the ratio 1 part milk to 19 parts black coffee.
You like your coffee to be as cool as possible when you drink it.
At what time (between t=0 and t=5) should you dilute the coffee with your milk? (Have a guess beforehand to see how good your intution is.)
Assumptions:
1. When you mix two substances of different temperatures T1 and T2, the resulting mixture has temperature the average of T1 and T2 weighted by proportion.
So in this case, if the coffee has temperature T1 and the milk has temperature T2 at the time of mixing, the mixture will have temperature 0.95T1 + 0.05T2.
2. Warm objects in a cool environment cool according to Newton's law of cooling:
dT/dt = -k(T-A), where A is the environment temperature and k is some unknown positive constant.
You have a boiling cup of coffee (so 100 degrees) and you wish to drink it in exactly 5 minutes time. You like to dilute your coffee with milk (at say 5 degrees) in the ratio 1 part milk to 19 parts black coffee.
You like your coffee to be as cool as possible when you drink it.
At what time (between t=0 and t=5) should you dilute the coffee with your milk? (Have a guess beforehand to see how good your intution is.)
Assumptions:
1. When you mix two substances of different temperatures T1 and T2, the resulting mixture has temperature the average of T1 and T2 weighted by proportion.
So in this case, if the coffee has temperature T1 and the milk has temperature T2 at the time of mixing, the mixture will have temperature 0.95T1 + 0.05T2.
2. Warm objects in a cool environment cool according to Newton's law of cooling:
dT/dt = -k(T-A), where A is the environment temperature and k is some unknown positive constant.
