1. A
cake is removed from the oven with an initial temperature of 450 degrees
Fahrenheit. It is set in a special room where it is allowed to cool. Ten minutes later, its temperature is
300 degrees. Twenty minutes after it was placed in the room, its temperature is
200 degrees. What is the temperature of the room? What kind of a room is it?
Use MAPLE to get a parametrized form of the solution to the differential equation
and move it to EXCEL
as follows. Create named variables corresponding to the parameters needed for
the fit.
Here these parameters should be 
Create a table of the available times
and temperatures for your problem. Add a third column that evaluates the
temperature T(t) using your formula from MAPLE in terms of the times in the
first column and the named parameters. Add a column of deviations squared and a
box for their sum. Finally, run solver to find the values of the parameters by
minimizing the sum squared error.
|
time |
Temperature |
T_MAPLE |
error^2 |
|
K |
|
|
0 |
450 |
=
Te+EXP(-K*A2)*(T0-Te) |
=(B2-C2)^2 |
|
Te |
|
|
10 |
300 |
=
Te+EXP(-K*A3)*(T0-Te) |
=(B3-C3)^2 |
|
T0 |
|
|
20 |
200 |
=
Te+EXP(-K*A4)*(T0-Te) |
=(B4-C4)^2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
=SUM(D2:D4) |
SSE |
|
|
2. The volume V of a sample of brewerÕs yeast similar to the one described in the online lecture on Integration by Substitution as a function of time t is given by
|
Time (hr) |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
Volume |
1.3 |
1.2 |
2.5 |
2.3 |
3.9 |
4.4 |
6.1 |
7.5 |
9.0 |
|
Time (hr) |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
|
Volume |
10.0 |
10.1 |
11.1 |
11.5 |
12.2 |
12.4 |
12.4 |
12.6 |
12.5 |
Assuming that the volume of this population grows according to the differential equation
and starts with 
, find the values of the parameters k, M,
and 
which give the best fit of the solution to the
differential equation in the least squares sense. Plot the data and the best
fit solution on the same set of axes.
Suggestion: Use MAPLE to find the solution to the differential equation in terms of the parameters and transfer the solution to EXCEL so you can use solver to minimize the sum squared error by adjusting the parameters. As usual, solver will need good guesses at starting values of the parameters to work from. You should be able to reduce the sum squared error to less than 1.5.
3. Write a paragraph describing to another person how to fit
a differential equation with unknown parameters to experimental data. Assume
that this person understands MAPLE and EXCEL commands but has not performed
this task before. Explain what is obtained at each step.