2. For many years, lead (Pb) was an additive to paint used to reduce molds and improve adhesion. Lead was also a gas additive used to improve combustion and reduce the knocking of car engines under stress. These sources have created a major problem of lead-laden dust, especially in the inner city and is problematic in the neural development of small children. Small children are exposed to lead through the dust ingested by normal hand-to-mouth play activities and from breathing the lead-laden dust. Once the lead enters the body it does not leave the body. This lead builds up in the children's bodies and has been linked to developmental problems with their nervous system. Scientists have discovered that lead concentrations as low as 10 mg/dl in the blood results in developmental toxicity.

a. The exposure of lead for children begins very low (since small babies hardly move), then increase to maximum during the early years from crawling and hand-to-mouth activities. As the child increases in height, he or she moves further away from the primary contaminated source and drops many of the hand-to-mouth activities, which lowers exposure. Assume that the weighted activity that exposes a boy to lead as a function of the age, t, in years satisfies the differential equation

A' = - kA + be-qt with A(0) = 0,

where A(t) is the activity time in hours per day. Suppose that the values of the parameters are k = 0.35 (yr-1), b = 6.0 (hr/day/yr), and q = 0.69 (yr-1). Solve this differential equation and graph the solution for 0 < t < 12. Find the maximum level of activity exposing the boy to lead and the age at which this occurs.

b. The lead enters the boy's body proportional to his weighted activity time, A(t), and does not leave following exposure. This suggests that the total amount of lead (P(t)) in his body (in mg) will satisfy the following differential equation:

P '(t) = KA(t) with P(0) = 0,

where A(t) is the solution obtained from Part a. and K = 500 micrograms-day/hour of play/yr. Find the solution P(t) and graph this solution for 0 < t < 12.

c. We have seen that the von Bertalanffy equation of growth provides an approximation for the weight gain of a child. Assume that the boy grows according to the initial value problem,

w'(t) = r(80 - w) with w(0) = 3.5,

where r = 0.053. Find the solution w(t) and graph this solution for 0 < t < 12. What would be the maximum weight of this boy for large values of t? (Note that the equation for w(t) loses accuracy significantly through the teenage years.)

d. Assume that this lead is uniformly distributed throughout the body. If the concentration of lead in the blood (in mg/dl), c(t), satisfies the equation,

c(t) = 0.1 P(t)/w(t),

then graph the solution for 0 < t < 12. Find the maximum concentration of lead in the boy and the age at which this occurs. Create a table for the boy at ages 2, 4, 8, and 12, giving the weighted activity level, A(t), the amount of lead in the body, P(t), the weight of the boy, w(t), and the concentration of lead in the body of the boy, c(t).

e. Check the website given above or any other sources and write a brief paragraph describing the risks that the boy modeled above might encounter.