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** of the species over one complete cycle (from \( t = 0 \) to \( t = 2\pi \)), we'll analyze the given population function: \[ P(t) = 200 + 50 \sin(t) \] ### Understanding the Function - **Baseline Population:** The constant term \( 200 \) represents the baseline population. - **Oscillation:** The term \( 50 \sin(t) \) models the fluctuation in population over time due to factors like birth rates, death rates, etc. ### Calculating Population Increases 1. **Identify When Population Increases:** - The population increases when the derivative \( P'(t) \) is positive. - Compute the derivative: \[ P'(t) = 50 \cos(t) \] - \( P'(t) > 0 \) when \( \cos(t) > 0 \), which occurs in the intervals: \[ 0 < t < \frac{\pi}{2} \quad \text{and} \quad \frac{3\pi}{2} < t < 2\pi \] 2. **Calculate the Increase Over Each Interval:** - **First Interval \([0, \frac{\pi}{2}]\):** \[ \int_{0}^{\frac{\pi}{2}} 50 \cos(t) \, dt = 50 \sin(t) \Big|_{0}^{\frac{\pi}{2}} = 50 (1 - 0) = 50 \] - **Second Interval \([\frac{3\pi}{2}, 2\pi]\):** \[ \int_{\frac{3\pi}{2}}^{2\pi} 50 \cos(t) \, dt = 50 \sin(t) \Big|_{\frac{3\pi}{2}}^{2\pi} = 50 (0 - (-1)) = 50 \] 3. **Total Population Increase:** \[ 50 + 50 = 100 \] ### Conclusion The **total population increase** over one complete cycle from \( t = 0 \) to \( t = 2\pi \) is **100** individuals. **Answer:** The total population increase over one complete cycle is 100 individuals.