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 $$ t=2\pi $$).

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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. The total population increase over one complete cycle is 100 individuals.

The population of a certain species in a protected area can be modeled by the function , where is measured in years. Determine the total population increase over one complete cycle (from to ).

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