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 behavior of the population function:

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

### Understanding the Function

- **Baseline Population:** The constant term $$ 200 $$ represents the average or baseline population.
- **Oscillation:** The term $$ 50 \sin(t) $$ causes the population to oscillate above and below this baseline.

### Key Points of the Cycle

1. **Maximum Population:**
   $$
   P_{\text{max}} = 200 + 50 = 250 \text{ individuals}
   $$
   This occurs when $$ \sin(t) = 1 $$, i.e., at $$ t = \frac{\pi}{2} $$.

2. **Minimum Population:**
   $$
   P_{\text{min}} = 200 - 50 = 150 \text{ individuals}
   $$
   This occurs when $$ \sin(t) = -1 $$, i.e., at $$ t = \frac{3\pi}{2} $$.

### Calculating Total Population Increase

The population increases in two distinct intervals within one cycle:

1. **First Increase:**
   - **From:** $$ t = 0 $$ ($$ P = 200 $$)
   - **To:** $$ t = \frac{\pi}{2} $$ ($$ P = 250 $$)
   - **Increase:** $$ 250 - 200 = 50 $$ individuals

2. **Second Increase:**
   - **From:** $$ t = \frac{3\pi}{2} $$ ($$ P = 150 $$)
   - **To:** $$ t = 2\pi $$ ($$ P = 200 $$)
   - **Increase:** $$ 200 - 150 = 50 $$ individuals

**Total Population Increase Over One Cycle:**
$$
50 \text{ (first increase)} + 50 \text{ (second increase)} = 100 \text{ individuals}
$$

### Conclusion

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

**Answer:**  
The total population increases by 100 individuals over one complete cycle.
The total population increases by 100 individuals over one complete cycle.

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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