As part of a liability defence (see the Wikipedia page onLiebeck v. McDonald's for a similar case), lawyers at Tim Hortonshave hired you to determine the temperature of a cup of TimHorton's coffee when it was initially poured. However, you onlyhave measurements of the coffee's temperature taken after it hasbeen purchased. According to Newton's Law of Cooling, an objectthat is warmer than a fixed environmental temperature will coolover 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 thetemperature of the object at time tt. The object has initialtemperature TinitTinit.

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

Transform the solution T(t)T(t) by putting the exponential term onone side and everything else on the other and taking natural logsof 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 uselinear least squares to estimate the unknown parameters TinitTinitand kk. Fit the transformed data to a line yi=b+axiyi=b+axi, i.e.,find the values of aa and bb which minimizef(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 thefollowing quantities:

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