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