Question

As part of a liability defence (see the Wikipedia page on Liebeck v. McDonald's for a similar case), lawyers at Tim Hortons have hired you to determine the temperature of a cup of Tim Horton's coffee when it was initially poured. However, you only have measurements of the coffee's temperature taken after it has been purchased. According to Newton's Law of Cooling, an object that is warmer than a fixed environmental temperature will cool over time according to the following relationship:

T(t)=E+(Tinit−E)e−ktT(t)=E+(Tinit−E)e−kt

where EE is the constant environmental temperature, and TT is the temperature of the object at time tt. The object has initial temperature TinitTinit.

Below you are given a data set measured from a purchased cup of coffee. The external temperature of the room is 2020 °C. The temperature of the coffee TiTi is given for several titi, where titi is the time in minutes since the coffee was poured.

Transform the solution T(t)T(t) by putting the exponential term on one side and everything else on the other and taking natural logs of both sides to get:

ln(T(t)−E)=ln(Tinit−E)−kt.ln⁡(T(t)−E)=ln⁡(Tinit−E)−kt.

Now transform the data below in the same way so that you can use linear least squares to estimate the unknown parameters TinitTinit and kk. Fit the transformed data to a line yi=b+axiyi=b+axi, i.e., find the values of aa and bb which minimize f(a,b)=∑i=1((yi)−(b+axi))2f(a,b)=∑i=1((yi)−(b+axi))2:

t_i (in minutes)	2	3	4	5	6	7	8	9	10
T_i (in °C)	86.1914	84.3832	88.5955	86.5824	86.7775	79.0971	80.4190	75.3221	74.7302

Use the computed coefficients aa and bb to calculate the following quantities:

What was the initial temperature TinitTinit of the coffee when it was poured?  °C
What is the time constant kk?  /min

Answer 1