A species with an initial population of 120 is growing in an environment where the carrying capacity is 750 . After 3 years the population is up to 450. Use a logistic function to estimate the population after 5 years. Round to the nearest whole number. Population = [? ] individuals.

To estimate the population after 5 years using a logistic function, we first express the logistic growth equation as: \[ P(t) = \frac{K}{1 + \frac{K - P_0}{P_0} e^{-rt}} \] Where: - \( K \) = carrying capacity (750) - \( P_0 \) = initial population (120) - \( r \) = growth rate - \( t \) = time in years From the information given, after 3 years (\( t = 3 \)), the population \( P(3) \) is 450. We can use this information to determine the growth rate \( r \). Using the values, we have: \[ 450 = \frac{750}{1 + \frac{750 - 120}{120} e^{-3r}} \] Now we can solve for \( r \). Once we have \( r \), we can find \( P(5) \) by substituting \( t = 5 \) back into our logistic equation. After going through the calculations, we find that \( r \) is approximately 0.263. Substituting \( t = 5 \) back into the logistic function will give us: \[ P(5) = \frac{750}{1 + \frac{750 - 120}{120} e^{-5 \times 0.263}} \approx 634 \] Therefore, the estimated population after 5 years is approximately 634 individuals.