Question

the logistic growth function $f(t) = \frac{117,000}{1 + 5100e^{-t}}$ 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?

a. the number of people initially infected is \boxed{}.
(round to the nearest whole number as needed.)

Explanation:

Step1: Identify initial time

At the start of the epidemic, \( t = 0 \). Substitute \( t = 0 \) into the logistic growth function \( f(t)=\frac{117000}{1 + 5100e^{-t}} \).

Step2: Substitute \( t = 0 \)

Since \( e^0 = 1 \), the function becomes \( f(0)=\frac{117000}{1 + 5100\times1}=\frac{117000}{5101} \).

Step3: Calculate the value

\( \frac{117000}{5101}\approx22.94 \), rounding to the nearest whole number gives 23.

Answer:

23