During a flu epidemic, the number of children in the Woodbridge Community School System who contracted influenza after t days was given by the following.
Q(t) = 8000/(1+199 e^(-0.6 t))
(a) How many children were stricken by the flu after the first day?
children
(b) How many children had the flu after 14 days?
children
(c) How many children eventually contracted the disease?
children
To find the number of children stricken by the flu after any given number of days, we need to substitute the value of 't' into the equation Q(t) = 8000/(1+199 e^(-0.6 t)).
(a) To find the number of children stricken by the flu after the first day (t = 1), we substitute t = 1 into the equation:
Q(t) = 8000/(1+199 e^(-0.6 t))
Q(1) = 8000/(1+199 e^(-0.6 * 1))
Now, we can calculate Q(1):
Q(1) = 8000/(1+199 e^(-0.6))
= 8000/(1+199 * e^(-0.6))
≈ 8000/(1+199 * 0.5488) [Since e^(-0.6) ≈ 0.5488]
≈ 8000/(1+109.2912)
≈ 8000/110.2912
≈ 72.57
So, after the first day, approximately 72.57 children were stricken by the flu.
(b) To find the number of children who had the flu after 14 days (t = 14), we substitute t = 14 into the equation:
Q(t) = 8000/(1+199 e^(-0.6 t))
Q(14) = 8000/(1+199 e^(-0.6 * 14))
Now, we can calculate Q(14):
Q(14) = 8000/(1+199 e^(-0.6 * 14))
= 8000/(1+199 * e^(-8.4))
≈ 8000/(1+199 * 0.00019) [Since e^(-8.4) ≈ 0.00019]
≈ 8000/(1+ 0.0378)
≈ 8000/1.0378
≈ 7707.10
So, after 14 days, approximately 7707.10 children had the flu.
(c) To find the number of children who eventually contracted the disease, we need to find the limit of Q(t) as t approaches infinity. We can use the formula:
lim(t->∞) Q(t) = lim(t->∞) 8000/(1+199 e^(-0.6 t))
As t goes to infinity, e^(-0.6t) approaches 0, so the denominator becomes 1. Therefore, the limit of Q(t) as t approaches infinity is:
lim(t->∞) Q(t) = lim(t->∞) 8000/1 = 8000
Therefore, eventually, approximately 8000 children contracted the disease.