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 need to analyze the population model:

$$
P(t) = 200 + 50 \sin(t)
$$

### Understanding the Model

- **Base Population:** The constant term $$ 200 $$ represents the base population around which the population fluctuates.
- **Oscillation:** The term $$ 50 \sin(t) $$ causes the population to oscillate above and below the base population.

### Analyzing the Increase

The **total population increase** refers to the sum of all increases in population during the cycle, not just the net change. Here's how to calculate it:

1. **Identify When Population Increases:**
   - The population increases when $$ \sin(t) $$ is increasing, which corresponds to intervals where the derivative $$ P'(t) $$ is positive.
   - The derivative is:
     $$
     P'(t) = 50 \cos(t)
     $$
   - $$ P'(t) > 0 $$ when $$ \cos(t) > 0 $$, which occurs in two intervals within one cycle:
     - **First Interval:** $$ 0 \leq t \leq \frac{\pi}{2} $$
     - **Second Interval:** $$ \frac{3\pi}{2} \leq t \leq 2\pi $$

2. **Calculate the Increase in Each Interval:**
   - **First Interval:**
     $$
     \int_{0}^{\frac{\pi}{2}} 50 \cos(t) \, dt = 50 \sin(t) \Big|_{0}^{\frac{\pi}{2}} = 50(1 - 0) = 50
     $$
   - **Second Interval:**
     $$
     \int_{\frac{3\pi}{2}}^{2\pi} 50 \cos(t) \, dt = 50 \sin(t) \Big|_{\frac{3\pi}{2}}^{2\pi} = 50(0 - (-1)) = 50
     $$

3. **Total Population Increase:**
   $$
   \text{Total Increase} = 50 + 50 = 100
   $$

### Conclusion

Over one complete cycle from $$ t = 0 $$ to $$ t = 2\pi $$, the **total population increase** is **100 individuals**.

**Answer:**  
The total population increase over one complete cycle is 100.
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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