Problem
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
Solution
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 \(\boxed{250}\) and the population size after 8 years is \(\boxed{1067}\).
