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. \[ P(t)=\frac{2800}{1+9 e^{-0.2 t}} \] Find the initial population size of the species and the population size after 7 years. Round your answers to the nearest whole number as necessary. \( \begin{array}{l}\text { Initial population size: } \\ \text { Population size after } 7 \text { years: } \square \text { fish }\end{array} \)

To determine the initial population size, we can evaluate the function \( P(t) \) at \( t = 0 \): \[ P(0) = \frac{2800}{1 + 9 e^{-0.2 \cdot 0}} = \frac{2800}{1 + 9 \cdot 1} = \frac{2800}{10} = 280 \] So, the initial population size is 280 fish. Next, we calculate the population size after 7 years by substituting \( t = 7 \): \[ P(7) = \frac{2800}{1 + 9 e^{-0.2 \cdot 7}} \approx \frac{2800}{1 + 9 e^{-1.4}} \] Calculating \( e^{-1.4} \approx 0.2466 \): \[ P(7) \approx \frac{2800}{1 + 9 \cdot 0.2466} = \frac{2800}{1 + 2.2194} \approx \frac{2800}{3.2194} \approx 869.66 \] Rounding to the nearest whole number, the population size after 7 years is approximately 870 fish. \[ \begin{array}{l} \text{Initial population size: 280 fish} \\ \text{Population size after 7 years: 870 fish} \end{array} \]