# Fourth laboratory - Fall 2002

## Warning

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).

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.

• Press the Invert button to see A-1. This requires the matrix to be nonsingular.
• Press the Eigen button to see the eigenvalues, with an eigenvector below each eigenvalue.
• Press the Solve button to see the solution of Ax=b. This requires the matrix to be nonsingular.
• Press the Multiply button to see the product Ab.
• You can keep three different matrices A, B, C and three different vectors a, b and c. To switch among them, click on the name to choose another one. The buttons always operate on the currently-chosen matrix and vector.
• The Copy button copies the latest result (if any) into the currently shown matrix or vector.
• An inverse matrix will be copied into the current matrix.
• For the result of Eigen, the three eigenvectors are copied into the columns of the current matrix. In the case of complex eigenvalues, the real part of a complex eigenvector is copied into one column and the imaginary part into the next column.
• For the result of Solve, the solution vector is copied into the current vector.

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?

• Using the eigenvalues and eigenvectors of A, find the general solution of the homogeneous system. Enter in Question 1 the eigenvalue closest to 0. Note how much closer to 0 this is than the other eigenvalues: the terms corresponding to the other eigenvalues will decay toward 0 quite quickly, but this one will decay very slowly.
• Suppose f(t) is the constant 1000. Find the steady-state solution by solving a system Ax = b. Enter in Question 2 the steady-state value of y1.
• Solve another system of equations to obtain the solution with initial condition corresponding to no lead in the body at time 0.
• Plot y1, y2 and y3 for t from 0 to 40, and find the values at t=40. Enter in Question 3 the value for y1 at t=40.
• What constant value of f(t) would have produced Mr. Jones's current value of y1= 2000? Enter this in Question 4.
• Using this f(t), what are the current values of y2 and y3? Enter the current value of y3 in Question 5.
• At what time t did y2 reach 90% of its current value? Enter this (to 3 decimal places) in Question 6.
• Find the solution starting with the current values at t=0, and assuming no further exposure to lead in the environment. Enter in Question 7 the amount of lead that will be in Mr. Jones's blood one year from now.
• Even with no further exposure to lead in the environment, y3 continues to increase for a while. Enter in Question 8 the value of t (to 3 decimal places) at which y3 reaches its maximum (considering the present as t=0).

## Typo!

Despite what it shows for Question 7 in the table above, do NOT work out y1(2) there. Instead give y1(1), as indicated in the story above.

## Submission

There is no reason in principle that you will not be able to submit your answers from home, but we cannot guarantee success. If a submission from within the Mathematics Department system is not successful, tell the TA.