Lead in the
Body
Part 4: A Lead-Free Environment
We suppose now that our
intrepid volunteer is placed in a totally lead-free environment after the
400 day exposure in Los Angeles. That is, the external driving force in
the model is now set to 0, so our new initial value problem is
y' = Ay,
y(400) = [x1(400), x2(400), x3(400)]T.
We have changed the name
of the dependent (vector) variable to y to avoid conflict with the
functions already defined, but we are leaving the time scale untouched
so that the x and y functions can be plotted together. That
is, the solution y is meaningful only for t at or beyond
400 days.
- You already know the general
solution of y' = Ay. What is it?
- Solve the initial value
problem stated above.
- Define the component functions
y1, y2, y3 for lead
levels in blood, tissue, and bone after 400 days. Plot these functions
for t from 400 days to 800 days. Describe what you
see.
- Combine your plots of x
and y functions. Interpret the combined plot in terms of lead levels
in the body.
- Your plot should show that
the lead level in the bones peaks at around 600 days. if the volunteer
remains in a lead-free environment, how long will it take for the lead
level in his bones to reach half of its peak amount?
