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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.

  1. You already know the general solution of y' = Ay. What is it?
  2. Solve the initial value problem stated above.
  3. 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.
  4. Combine your plots of x and y functions. Interpret the combined plot in terms of lead levels in the body.
  5. 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?
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