The logistic growth function f(t)= 100,000/5000e^-t describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a particular community.
A) How many people became ill with the flu after the epidemic began?
B) What is the limiting size of the population that becomes ill?
I think there's a typo. As written it is just
20 e^-t
Anyway, since I have no idea when the plague began, I have no value for t.
As for B, the typo does not allow for a limiting population.
A) To find out how many people became ill with the flu after the epidemic began, we need to find the value of f(t) when t = 0 (since t is measured in weeks after the initial outbreak).
Substituting t = 0 into the logistic growth function:
f(0) = 100,000 / (5000 * e^0)
f(0) = 100,000 / (5000 * 1) (since e^0 = 1)
f(0) = 20
Therefore, 20 people became ill with the flu after the epidemic began.
B) The limiting size of the population that becomes ill is the maximum number of people who can become ill with the flu in this particular community.
In the logistic growth function, the limiting size of the population is represented by the constant term, which is given as 100,000 in this case.
Therefore, the limiting size of the population that becomes ill is 100,000.
A) To find how many people became ill with the flu after the epidemic began, we need to evaluate the function at t = 0.
f(t) = 100,000/(5000e^(-t))
Substituting t = 0:
f(0) = 100,000/(5000e^(-0))
= 100,000/(5000 * 1)
= 100,000/5000
= 20
Therefore, 20 people became ill with the flu after the epidemic began.
B) The limiting size of the population that becomes ill can be found by taking the limit of the function as t approaches infinity:
lim(t→∞) f(t) = lim(t→∞) (100,000/(5000e^(-t)))
As t approaches infinity, the term e^(-t) approaches 0, making the denominator very large. So, the limit will approach 0.
lim(t→∞) f(t) = 100,000/(5000 * 0)
= 100,000/0
The division by 0 is undefined, so the limiting size of the population that becomes ill is infinite. In practical terms, it means that the number of people getting the flu can keep increasing without an upper bound.