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 determine the total population increase over one complete cycle from \( t=0 \) to \( t=2\pi \), we need to calculate the population at the beginning and the end of the cycle. 1. **At \( t=0 \)**: \[ P(0) = 200 + 50 \sin(0) = 200 + 50 \cdot 0 = 200 \] 2. **At \( t=2\pi \)**: \[ P(2\pi) = 200 + 50 \sin(2\pi) = 200 + 50 \cdot 0 = 200 \] Now, we calculate the total population increase over one complete cycle: \[ \text{Population Increase} = P(2\pi) - P(0) = 200 - 200 = 0 \] Thus, the total population increase over one complete cycle is **0**. Now, here's some additional fun information! The sine function, as part of our population model, dances between -1 and 1. This means during its peak, the population reaches 250, while in its trough, it dips to 150, illustrating nature's ebb and flow. Talk about a wild rollercoaster ride of numbers! In real-world scenarios, such population functions can help biologists assess species' health in conservation areas. By understanding population dynamics, conservation efforts can be tailored effectively, ensuring that the ups and downs of species populations are managed for sustainability!