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 evaluate the function \( P(t) = 200 + 50 \sin(t) \) at the endpoints of the interval. At \( t = 0 \): \[ P(0) = 200 + 50 \sin(0) = 200 + 50 \cdot 0 = 200 \] At \( t = 2\pi \): \[ P(2\pi) = 200 + 50 \sin(2\pi) = 200 + 50 \cdot 0 = 200 \] Now, to determine the total increase, we subtract the population at the start from the population at the end: \[ \text{Total increase} = P(2\pi) - P(0) = 200 - 200 = 0 \] Thus, the total population increase over one complete cycle is \( 0 \). The population oscillates between a maximum of \( 250 \) (when \( \sin(t) = 1 \)) and a minimum of \( 150 \) (when \( \sin(t) = -1 \)), but since it returns to the same value after each cycle, there's no net increase.