Notebook to support Exercise 3.4.11, modeling a cooling potato.

The data, in time/temperature pairs:

data = {{0, 204}, {2, 193}, {4, 184}, {8, 169}, {10, 162}, {13, 156}, {17, 149}, {20, 143}, {24, 138}, {30, 130}}

The number of data points is

n = Length[data]

A plot:

plt1=ListPlot[data,AxesLabel->{"time (minutes)","Temperature"},PlotStyle->{Red}]

We seek to fit a function

A = 72; u0 = 204;
u[t_] = A + ((u0-A)^(1-r) + k*(r-1)*t)^(1/(1-r))

A least-squares function can be formed as

SS = Sum[(u[data[[j,1]]]-data[[j,2]])^2,{j,1,n}]

Now adjust k and r to minimize this.

Plot3D[Log[SS],{k,0,0.0004},{r,2,2.5}]