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Use a graphing calculator and the following scenario:

The population [tex]\( P \)[/tex] of a fish farm in [tex]\( t \)[/tex] years is modeled by the equation
[tex]\[ P(t) = \frac{2800}{1 + 9 e^{-0.8 t}}. \][/tex]

What is the initial population of fish?

[tex]\[ \square \text{ fish} \][/tex]

Sagot :

GinnyAnswer
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.
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