Let's analyze the given logistic growth function:

$f(t) = \frac{104,000}{1 + 5800e^{-t}}$

Part (a): How many people became ill with the flu when the epidemic began?

At the beginning of the epidemic, $t = 0$. We can substitute $t = 0$ into the logistic growth function to find the initial number of people infected:

$f(0) = \frac{104,000}{1 + 5800e^{0}} = \frac{104,000}{1 + 5800 \cdot 1} = \frac{104,000}{5801}$

Calculating this:

$f(0) \approx \frac{104,000}{5801} \approx 17.93$

So, the number of people initially infected is approximately 18 people.

Part (b): How many people were ill by the end of the fourth week?

Now, we substitute $t = 4$ into the logistic growth function to calculate the number of people infected after 4 weeks:

$f(4) = \frac{104,000}{1 + 5800e^{-4}}$

We need to evaluate $e^{-4}$:

$e^{-4} \approx 0.01832$

Substitute this back into the equation:

$f(4) = \frac{104,000}{1 + 5800 \times 0.01832} = \frac{104,000}{1 + 106.256} = \frac{104,000}{107.256} \approx 969.51$

So, approximately 970 people were infected by the end of the fourth week.

Part (c): What is the limiting size of the population that becomes ill?

The limiting size of the population corresponds to the carrying capacity, which is the maximum value $f(t)$ approaches as $t \to \infty$. For logistic growth functions, the limiting value is the numerator of the function, which in this case is:

$\lim_{t \to \infty} f(t) = 104,000$

So, the limiting size of the population that becomes ill is 104,000 people.

Would you like more details or have any questions about these calculations?

Here are five related questions you can explore:

1. How would the model change if the initial number of infected individuals was larger?
2. What happens to the infection rate if the constant 5800 changes?
3. How does the infection spread in the early versus later weeks based on this logistic model?
4. What are the implications of the limiting size in real-life epidemic control strategies?
5. Can this model be applied to other diseases or conditions? If so, how?

Tip: Logistic functions are commonly used to model population growth, disease spread, and resource-limited growth because they describe systems with an initial exponential phase followed by a leveling off due to constraints.