Math 3280 - Hampton - Lab 2 -  Due 2/2

I measured the temperature (in Fahrenheit) of a cup of tea at five different times, shown in the chart below.

Time 		Temp
10:30 		175
10:33 		163
10:37		152
10:42		142
10:52		127

In this lab you will use Newton' s law of cooling, dT/dt = k*(A-T), to explore modeling this data.  As in class, T denotes the temperatue, A is the ambient temperature, and t is the time.  It is simplest to let t = 0 correspond to 10:30 am.Assume that the ambient temperature A was 70 F.

Exercises

1. Use the first two data points to calculate k.  Plot the solution to the ODE with the data - does it look correct?

2. Use the last two data points to calculate k.  Again, plot the solution along with the data.  Does it look better or worse?

3. There is always some error in experimental data.  When determining k from the first two data points, is the value of k more sensitive to a 10% error in the time or temperature of the second data point?

4. Use the interactive cell below to find a good guess for k.  Which data point do you think has the largest error in temperature?

(* For exercise 4 *)
tdata = {{0,175},{3,163},{7,152},{12,142},{22,127}};
Manipulate[Show[{ListPlot[tdata,PlotStyle->{Red,PointSize[.015]}],Plot[70+(T0-70)*Exp[-(t-t0)*k],{t,0,22}]},PlotRange->{{0,22},{120,185}}],{{k,.03},.01,.05,.001},{{T0,175},120,185,.1},{{t0,0},0,22,.1}]