A cup of coffee cools according to Newton’s law of
cooling (see below). Use data from the graph of the temperature
T(t) in Figure 1.3.9 to estimate the constants Tm, T0, and
k in a model of the form of a first-order initial-value
problem:
$$dT/dt = k(T – Tm),T(0) = T0
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$$
I was able to estimate from the graph that Tm=75 and T0=180, and I understand that you can solve for k but I don't see the book found dT/dt approximately equal to 1.
Books answer:
From the graph in the text we estimate To = 180° and Tm = 75°. We observe that when T = 85, $$dT/dt \approx – 1
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$$
$$k = \frac{{dT/dt}}{{T – Tm}} = \frac{{ – 1}}{{85 – 75}} = – 0.1
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$$
Did they determine this by estimating the slope or am I missing something?
Best Answer
The book's answer is quite mystifying and possibly wrong.
Finding the point on the graph where $dT/dt = -1$ by eye is extremely hard, especially because the two axes have different scales, so you're not looking simply for a place in the figure where the slope is $45^\circ$ visually. And even if you are looking for the right slope, the whole computation hinges on estimating its distance from $T_m$ to be one fifth of the smallest gradation on the $y$ axis!
It would be much more reliable to estimate the slope at $t=0$ (by drawing a tangent at that point and finding its $x$-intercept, then dividing $180$ by that), and then solve for $k$ using that estimate.