The logistic growth function $f(t)=\frac{105,000}{1+4600e^{-t}}$ 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 when the epidemic began?
b. How many people were ill by the end of the fourth week?
c. What is the limiting size of the population that becomes ill?

Step 1 ：The logistic growth function $f(t)=\frac{105,000}{1+4600e^{-t}}$ describes the number of people, $f(t)$, who have become ill with influenza t weeks after its initial outbreak in a particular community.

Step 2 ：For part a, we need to find the value of the function at t=0. This will give us the number of people who became ill when the epidemic began.

Step 3 ：Substituting t=0 into the function, we get approximately $23$ people.

Step 4 ：For part b, we need to find the value of the function at t=4. This will give us the number of people who were ill by the end of the fourth week.

Step 5 ：Substituting t=4 into the function, we get approximately $1232$ people.

Step 6 ：For part c, we need to find the limit of the function as t approaches infinity. This will give us the limiting size of the population that becomes ill.

Step 7 ：As t approaches infinity, the function approaches $105,000$.

Step 8 ：Final Answer: a. The number of people who became ill when the epidemic began is approximately $\boxed{{\displaystyle 23}}$. b. The number of people who were ill by the end of the fourth week is approximately $\boxed{{\displaystyle 1232}}$. c. The limiting size of the population that becomes ill is $\boxed{{\displaystyle 105,000}}$.