To determine the initial population of fish, we need to evaluate the population function [tex]\( P(t) = \frac{2800}{1 + 9 e^{-0.8 t}} \)[/tex] at [tex]\( t = 0 \)[/tex].
1. Substitute [tex]\( t = 0 \)[/tex] into the function.
[tex]\[
P(0) = \frac{2800}{1 + 9 e^{-0.8 \cdot 0}}
\][/tex]
2. Simplify the exponent.
Since [tex]\( -0.8 \cdot 0 = 0 \)[/tex], we have:
[tex]\[
e^{0} = 1
\][/tex]
3. Substitute [tex]\( e^{0} = 1 \)[/tex] back into the equation.
[tex]\[
P(0) = \frac{2800}{1 + 9 \cdot 1}
\][/tex]
4. Simplify the denominator.
[tex]\[
1 + 9 \cdot 1 = 1 + 9 = 10
\][/tex]
5. Calculate the fraction.
[tex]\[
P(0) = \frac{2800}{10} = 280
\][/tex]
Therefore, the initial population of fish is [tex]\( \boxed{280} \)[/tex] fish.
