The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)).

To find the total population increase over one complete cycle from \( t = 0 \) to \( t = 2\pi \), we first evaluate the function \( P(t) = 200 + 50 \sin(t) \) at the endpoints of the interval. 1. **At \( t = 0 \)**: \[ P(0) = 200 + 50 \sin(0) = 200 + 50 \cdot 0 = 200 \] 2. **At \( t = 2\pi \)**: \[ P(2\pi) = 200 + 50 \sin(2\pi) = 200 + 50 \cdot 0 = 200 \] The population at both \( t = 0 \) and \( t = 2\pi \) is 200, which indicates there is no overall change in population. However, to determine the total population increase over one complete cycle, we consider the maximum and minimum values of \( P(t) \) during the cycle: - The sine function \( \sin(t) \) reaches its maximum value of 1 and minimum value of -1. - Maximum population: \[ P_{\text{max}} = 200 + 50 \cdot 1 = 250 \] - Minimum population: \[ P_{\text{min}} = 200 + 50 \cdot (-1) = 150 \] The total population increase within the cycle can be calculated as the difference between the maximum and minimum values: \[ \text{Total Increase} = P_{\text{max}} - P_{\text{min}} = 250 - 150 = 100 \] Thus, the total population increase over one complete cycle is \( \boxed{100} \).