To determine how many people became ill with the virus when the epidemic began, we need to evaluate the given function [tex]\( N(t) \)[/tex] at the point [tex]\( t = 0 \)[/tex].
Given the function:
[tex]\[
N(t) = \frac{30,000}{1 + 40e^{-2.0t}}
\][/tex]
we substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
N(0) = \frac{30,000}{1 + 40e^{-2.0 \cdot 0}}
\][/tex]
Since [tex]\( e^0 = 1 \)[/tex], we get:
[tex]\[
N(0) = \frac{30,000}{1 + 40 \cdot 1}
\][/tex]
Simplify the denominator:
[tex]\[
N(0) = \frac{30,000}{1 + 40} = \frac{30,000}{41}
\][/tex]
By evaluating this expression, we find the initial number of people who were ill with the virus:
[tex]\[
N(0) \approx 732
\][/tex]
Thus, when the epidemic began, approximately 732 people were ill with the virus.
