As children age, their play activity generally decreases their exposure to lead
ID: 2855110 • Letter: A
Question
As children age, their play activity generally decreases their exposure to lead (though initially it increases as a baby goes from being immobile through crawling). An alternative model, which shows lead concentrations in children as they age, is given below. (It is inappropriate for young children 0-2.)
A.) Assume that the weight (in kg) of a girl satisfies the differential equation
dw/dt = 0.07(65 w), w(0) = 3
where t is in years. Solve this differential equation. Find the predicted weights for a girl ages 2, 4, 7, and 10.
B.) Because of the declining exposure, the amount of lead in the body, P (in g), satisfies the differential equation
dP(t)/ dt = 100e^(0.4t)*w(t), P (0) = 0
where w(t) is the solution from Part a. Solve this differential equation and determine the amount of lead in a girl at ages 2, 4, 7, and 10.
C.) If the lead is uniformly distributed in the body, then the concentration of lead in the blood, c (in g/dl), satisfies the equation,
c(t) = 0.1P(t)/ w(t)
Find the concentration of lead in a girl ages 2, 4, 7, and 10.
Explanation / Answer
dw/dt = 0.07(65-w)
Rearranging it,
dw/(65-w) = 0.07 dt
on integrating it,
ln(65 - w) = 0.07t + c
w = ce^(-0.07t) + 65
To find c, put t=0
w(0) = 3
3 = c + 65
c = -62
Using this,
w = -62e^(-0.07t) + 65
at t=2,
w = -62*e^(-0.14) + 65
w = 11.09
at t=4
w = -62*e^(-0.28) + 65
w = 18.14
at t=7,
w = -62e^(-0.49) + 65
w = 27.017
at t=10,
w = -62e^(-0.7) + 65
w = 34.21
Similarly we can solve b and c
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