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Fourth laboratory - Fall 2002

Due: Friday, November 1, 2001 at 11:30pm

Warning

For technical reasons, while working on this lab, you must not reload or leave this page within your web browser. If you do leave and want to come back again, you should quit your browser and restart it, either from the background menu or a terminal window. Also, save your answers frequently by using the submittor at the end of the page. The section of the text relevant to this lab is 10.4. Note the following useful facts:

1. If A is a nonsingular matrix and g is a constant vector, the steady-state solution of the system x' = A x + g is x = -A-1 g.

2. The general solution of the system x' = A x + g can be written

x(p)(t) + c1 x(1)(t) + ... + cn x(n)(t) = x(p)(t) + P(t) c

where x(p)(t) is a particular solution and P(t) is the matrix whose columns are the fundamental set of solutions x(1)(t), ... x(n)(t) of the homogeneous system. Thus the coefficients in the solution for the initial condition x(t0) = x(0) can be obtained by solving the system P(t0) c = x(0) - x(p)(t0).

Lead in the Body

When lead enters the human body (whether in food, air or water) it first goes into the blood, and from there to the bones and teeth, where it can remain for a long time, and the soft tissues where it can cause damage. This may be modeled by a "three-compartment model" involving three dependent variables representing the amounts of lead in the three compartments: blood (y1), soft tissues (y2), and bones (y3). The rate at which lead transfers from compartment i to compartment j is proportional to the amount yi in compartment i, with a coefficient denoted by aij. There is no direct transfer between the bones and soft tissues, so a23 and a32 are zero. There is also transfer from the blood and soft tissues to the external environment through urine, hair, nails and sweat, corresponding to coefficients a10 and a20. If f(t) is the rate at which lead enters the blood from the environment, we get the 3 x 3 linear system

We measure time in units of years, and amounts of lead in micrograms. The rate constants for a healthy adult have been estimated as follows:
a10=7.70 a12=4.05 a13=1.42
a20=5.92 a21=4.53
a31=0.0128

Mr. Jones has just retired from his job as a factory worker, in which for 40 years he was constantly exposed to high lead levels. He now has 2000 micrograms of lead in his blood. He will not have any significant exposure to environmental lead in the future, but the lead in his bones is still a concern.

This lab uses two Java applet tools: a matrix calculator and a solution grapher.

The matrix calculator below computes inverses, eigenvalues and eigenvectors of 3 x 3 matrices, multiplies a matrix and a vector, and solves the matrix-vector equation Ax = b. You can change the entries in the matrix A and vector b by clicking on them and typing.

The plotter below can be used to plot any function of the form y = a0+a1 exp(r1t) +a2 exp(r2t) +a3 exp(r3t) or its derivative. Enter the coefficients, enter the initial and final values of t, and click the "Plot y" or "Plot y'" button. The values of y (or y') for t running between the initial and final values are shown. If you click on the graph, the coordinates of the point where you clicked are shown near the top left of the graph.

How much lead did Mr. Jones accumulate in his bones?