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Newton s Law of Cooling problem?
Newton s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton s law of cooling.
If the coffee has a temperature of 190 degrees Fahrenheit when freshly poured, and 2 minutes later has cooled to 160 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 142 degrees.
When will the coffee reach a temperature of 142 degrees?
2 Answers
- ?Lv 76 years agoFavorite Answer
In mathematical terms Newton's law of cooling says:
dT/dt = -k ( T - Ts),
where k is a constant and Ts is the surrounding temperature.
This differential equation has the solution:
T(t) = Ts + ( T(0) - Ts ) exp(-k t)
What this says is that the initial temperature difference decays away exponentially. From the data we have
t=2:
160 = 64 + 126 exp(-2k)
exp(2k) = 126/96
2 k = ln(126/96)
k = 1/2 * ln(126/96) = 0.136
Having obtained k we can solve
142 = 64 + 126 exp(-0.136 t)
0.136 t = ln(126/78)
t = 3.5 minutes
- Anonymous6 years ago
[T1(V1)]/t1=[T2(V2)]/t2
T=temperature, F
t= time, minute
V= volume
but if volume of coffee remains constant, then V1=V2.
T1/t1 = T2/t2
(190-160)/2 = (160-142)/t2
30/2 = 18/t2
t2 = 18(2) /30
t2 = 1.2 minutes
Therefore, the coffee reaches a temp. of 142F after 1.2 minutes.