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. \( \begin{array}{l}\text { Initial population size: } \\ \text { Population size after } 7 \text { years: } \square \text { fish }\end{array} \)

Substitute \( t=0 \) into the expression \( \frac{2800}{1+9 e^{-0.2 t}} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{2800}{\left(1+9exp\left(-0.2t\right)\right)}\) - step1: Substitute: \(\frac{2800}{\left(1+9exp\left(-0.2\times 0\right)\right)}\) - step2: Multiply: \(\frac{2800}{\left(1+9exp\times 0\right)}\) - step3: Multiply: \(\frac{2800}{\left(1+0\right)}\) - step4: Remove 0: \(\frac{2800}{1}\) - step5: Divide the terms: \(2800\) Substitute \( t=7 \) into the expression \( \frac{2800}{1+9 e^{-0.2 t}} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{2800}{\left(1+9exp\left(-0.2t\right)\right)}\) - step1: Substitute: \(\frac{2800}{\left(1+9exp\left(-0.2\times 7\right)\right)}\) - step2: Multiply the numbers: \(\frac{2800}{\left(1+9exp\left(-1.4\right)\right)}\) - step3: Multiply: \(\frac{2800}{\left(1-12.6exp\right)}\) - step4: Remove the parentheses: \(\frac{2800}{1-12.6exp}\) - step5: Simplify the expression: \(\frac{14000}{5-63exp}\) Calculate \( \frac{14000}{5-63exp} \). Simplify the expression by following steps: - step0: Solution: \(\frac{14000}{5-63exp}\) The initial population size of the species is 2800 fish. To find the population size after 7 years, we need to calculate \( \frac{14000}{5-63e^{-1.4}} \). Calculate the value by following steps: - step0: Calculate: \(\frac{14000}{5-63e^{-1.4}}\) - step1: Convert the expressions: \(\frac{14000}{5-63e^{-\frac{7}{5}}}\) - step2: Multiply the terms: \(\frac{14000}{5-\frac{63e^{\frac{3}{5}}}{e^{2}}}\) - step3: Subtract the numbers: \(\frac{14000}{\frac{5e^{2}-63e^{\frac{3}{5}}}{e^{2}}}\) - step4: Multiply by the reciprocal: \(14000\times \frac{e^{2}}{5e^{2}-63e^{\frac{3}{5}}}\) - step5: Multiply: \(\frac{14000e^{2}}{5e^{2}-63e^{\frac{3}{5}}}\) - step6: Simplify: \(\frac{14000e^{2}}{5e^{2}-63\sqrt[5]{e^{3}}}\) The population size after 7 years is approximately 1329 fish.