(5 points) 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) where E is the constant environmental temperature, and T is the temperature of the object at time t. The object has initial temperature Tnit . Below you are given a data set measured from a purchased cup of coffee. The external temperature of the room is 20 *C. The temperature of the coffee T, is given for several ti, where t, is the time in minutes since the coffee was poured. Transform the solution T) by putting the exponential term on one side and everything else on the other and taking natural logs of both sides to get In(T(t) _ E) = In(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 Tinit and k. Fit the transformed data to a line yi = b + axi, i.e., find the values of a and b which minimize f(a, b) = Σ! ((yi)-(b + ari))2: t i (in minutes 2 4 6 10 TI (in °c) 92.4450 86.6854 89.8068 81.6783 81.901581.3594 81.9200 80.9616 75.1080 Use the computed coefficients a and b to calculate the following quantities: What was the initial temperature Tini of the coffee when it was poured? "C What is the time constant k? /min
