A species of fish was added to a lake. The population size $P (t)$ of this species can be modeled by the following function, where $t$ is the number of years from the time the species was added to the lake.
$P (t) = \frac{2000}{1 + 7 e^{- 0.26 t}}$
Find the initial population size of the species and the population size after 8 years. Round your answers to the nearest whole number as necessary.
Initial population size:
Population size after 8 years:
fish
fish

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Answer

Final Answer: The initial population size of the species is $250$ and the population size after 8 years is $1067$ .

Steps

Step 1 ：The initial population size can be found by substituting $t = 0$ into the function $P (t)$ .

Step 2 ： $P (0) = \frac{2000}{1 + 7 e^{- 0.26 \times 0}} = 250$ fish

Step 3 ：The population size after 8 years can be found by substituting $t = 8$ into the function $P (t)$ .

Step 4 ： $P (8) = \frac{2000}{1 + 7 e^{- 0.26 \times 8}} = 1067$ fish

Step 5 ：Final Answer: The initial population size of the species is $250$ and the population size after 8 years is $1067$ .