Problem set #5

The following is a true story. My neighbor, Mrs. X [not her real name, of course], approached me for some help in a lawsuit which she had brought against her former employers. She had suffered serious health problems and attributed them to a faulty furnace at her former place of employment. The furnace had cracked and was leaking carbon monoxide (CO) into the building. Continual exposure to carbon monoxide can cause a wide variety of health problems. An environmental consulting firm hired by the defendants measured the extent of the leak in the following way. They turned off the furnace and allowed the building to vent for many hours.

 

Table 1: Tables of first (left) and second (right) carbon monoxide measurements.

$\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 60 & 19.1 \\ \hline 70 & 19.8 \\ \hline 80 & 20.5 \\ \hline \end{tabular}}$ $\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 50 & 21.9 \\ \hline 60 & 22.2 \\ \hline 70 & 22.3 \\ \hline \end{tabular}}$

Then, they turned on the furnace and measured the carbon monoxide levels at a duct to a common area at regular time intervals (see Table 1). After this test, they turned off the furnace, waited a few hours, and ran a second test (see Table 1 again).

1.

Develop a mathematical model for the carbon monoxide level at the end of the duct assuming that the faulty furnace was faulty and is emitting carbon monoxide. What are the unknown parameters in your model?

2.

Fit your model to the measured data. Describe your procedure explicitly. What is the difference between your model predictions and the raw data? Extrapolate your solution out to a time of 240 minutes.

3.

If the measured data were subjected to small uniform random fluctuations, how would it impact your model fit? Justify your answer.