Questions: Evaluating an exponential function with base e that models a real-world... 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) = 1200 / (1 + 2 e^(-0.3 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: fish Population size after 8 years: fish

Solution Steps

To find the initial population size, evaluate the function $P(t)$ at $t = 0$. For the population size after 8 years, evaluate the function at $t = 8$. Use Python to compute these values and round the results to the nearest whole number.

To find the initial population size, we evaluate the function $P(t)$ at $t = 0$: $$P(0) = \frac{1200}{1 + 2 e^{-0.3 \cdot 0}} = \frac{1200}{1 + 2 \cdot 1} = \frac{1200}{3} = 400$$ Thus, the initial population size is $400$ fish.

Next, we evaluate the function $P(t)$ at $t = 8$: $$P(8) = \frac{1200}{1 + 2 e^{-0.3 \cdot 8}} = \frac{1200}{1 + 2 e^{-2.4}}$$ Calculating $e^{-2.4}$ gives approximately $0.090717953$: $$P(8) = \frac{1200}{1 + 2 \cdot 0.090717953} = \frac{1200}{1 + 0.181435906} = \frac{1200}{1.181435906} \approx 1016$$ Thus, the population size after 8 years is $1016$ fish.

Final Answer