[ Mathematics 256 home page ]
Fourth laboratory  Fall 2002
Due: Friday, November 1, 2001 at 11:30pm
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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 steadystate 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) + c_{1} x^{(1)}(t)
+ ... + c_{n} 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(t_{0}) = x^{(0)} can be obtained
by solving the system P(t_{0}) c = x^{(0)}
 x^{(p)}(t_{0}).
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
"threecompartment model" involving three dependent variables
representing the amounts of lead in the three compartments:
blood (y_{1}),
soft tissues (y_{2}), and
bones (y_{3}). The rate at which lead transfers
from compartment i to compartment j is proportional
to the amount y_{i} in compartment i, with
a coefficient denoted by a_{ij}. There is no
direct transfer between the bones and soft tissues, so
a_{23} and a_{32} 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
a_{10} and a_{20}. 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:
a_{10}=7.70 
a_{12}=4.05 
a_{13}=1.42 
a_{20}=5.92 
a_{21}=4.53 
a_{31}=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
matrixvector 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 currentlychosen 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 =
a_{0}+a_{1} exp(r_{1}t)
+a_{2} exp(r_{2}t)
+a_{3} exp(r_{3}t) 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 steadystate solution by solving a system
Ax = b. Enter in Question 2 the steadystate
value of y_{1}.
 Solve another system of equations to obtain the solution with initial
condition corresponding to no lead in the body at time 0.
 Plot y_{1}, y_{2} and
y_{3} for t from
0 to 40, and find the values at t=40. Enter in Question 3 the
value for y_{1} at t=40.
 What constant value of f(t) would have produced Mr. Jones's
current value of y_{1}= 2000? Enter this in Question 4.
 Using this f(t), what are the current values of
y_{2} and y_{3}? Enter the current value
of y_{3} in Question 5.
 At what time t did y_{2} 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,
y_{3} continues to increase for a while.
Enter in Question 8 the value of t (to 3 decimal places)
at which
y_{3} 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
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Send mathematical questions about the labs to
the professor in charge of Math 256:
fournier@math.ubc.ca