Coffee in a cup cools down according to Newton's Law of Cooling:

dT/dt = k(T - T_m) where k is a constant of proportionality. (Note: if T_m is constant, and since the cup is cooling (that is, T > T_m), the constant k < 0.) Use data from the graph below which is of the temperature to estimate T_m, T_0, and k in a model of the form above (that is, dT/dt = k(T - T_m), T(0) = T_0.

Well, the graph starts at around 175 on the T axis (y-axis) and falls exponentially down until leveling off at around T = 80 to a horizontal line. Note that the T axis goes up in increments of 50.

The x-axis, that is the minutes, goes from 0 - 100, in increments of 25.

The points:

(0, 175)
(25, 88ish)
(50, 80ish)
(75, 80ish)
(100, 80ish)

You have a slight typo in your equation:
In the is the "bath" temperature: the temperature of the surroundings. This equation has the solution:

where , the temperature of the substance at the instant it starts to cool. (If the coefficient on the T in your differential equation is positive then we have a positive coefficient in the exponent, which implies that the temperature not only increases, but becomes arbitrarily large as t increases, which is a ridiculous assumption about the cooling!)

Note that as time goes to infinity the temperature of the substance becomes . So you know and . So just pick an arbitrary time (one where you can read the coordinate easily) and solve for k:


-Dan

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