Calculus is COOL!
In class, we’ve seen that Newton’s Law of Cooling can be expressed in the form of an ODE. Namely, the rate of change of the temperature of an object is proportional to the temperature difference between the environment and the object, that is
\[T'(t)=r[T_{\text{env}}-T(t)]\]
Here \(r\) is a constant (the coefficient of heat transfer) and \(T_{\text{env}}\) is the temperature of the environment (also assumed to be constant).
By solving the initial-value problem using separation of variables we have obtained
\[\begin{equation} T(t)=T_{\text{env}}+(T_0-T_{\text{env}})e^{-rt} \tag{1} \end{equation}\]
where \(T_0=T(0)\) is the initial temperature of the object.
Experiment 1. During the lecture, we’ve done a 10-minute experiment with a mystery liquid [hot water]. At \(t=0\), we measured \(T_{\text{env}}=21.3^\circ\text{C}\), \(T_0=50^\circ\text{C}\) and after \(5\) minutes we measured \(T_5=47^\circ\text{C}\). These data, when compared to the solution in eq. (1), allowed us to find a value for \(r\):
\[r=\frac{1}{5}\ln\left(\dfrac{T_0-T_{\text{env}}}{T_5-T_{\text{env}}}\right)\approx 0.02208\]
Thus we were able to predict the temperature at \(t=10\) (minutes):
\[T(10)=T_{\text{env}}+(T_0-T_{\text{env}})e^{-10r}\approx 44.3^\circ\text{C}\]
and our (rather primitive) thermometer read \(T_{10}=44^\circ\text{C}\).
"Math is a wonderful thing! Math is a really cool thing!"
Below you can see the data points recorded by the thermometer:
Figure 1: Data from the 10-minute experiment we did between 9:22-9:32 on Thursday [Oct 20].
Experiment 2. And here’s the data from a 90-minute experiment I’ve done the day before. \(T_{\text{env}}=20.5^\circ\text{C}\), \(T_0=78^\circ\text{C}\), \(T_{10}=63^\circ\text{C}\).
Figure 2: Data from a 90-minute experiment done on Wednesday [Oct 19].
Questions:
What is the value of the coefficient \(r\) in Experiment 2?
How good is the prediction of the solution \(T(t)\) for the measured temperatures \(T_{20}\), \(T_{40}\), \(T_{60}\), \(T_{80}\)?
Fit a function of the form \(T_{\text{env}}^\ast+(T_0-T_{\text{env}}^\ast)e^{-r^\ast t}\) to the data using the data points \(T_0\), \(T_{10}\), \(T_{20}\), \(T_{30}\). What values does your fit assign to \(T_{\text{env}}^\ast\) and \(r^\ast\)? Are these value close to \(T_{\text{env}}\) and \(r\)?
Finally, here’s a related problem: Suppose you have a cup of hot coffee and some cream at room temperature. You need to go make a phone call for 5 minutes. Should you mix in the cream before or after the call to ensure your coffee stays as hot as possible?
[SPOILER ALERT] A video of Matt Parker explaining and demonstrating what will happen: