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 over one complete cycle of the function $$ P(t) = 200 + 50 \sin(t) $$, we will follow these steps:

1. **Identify the function and its parameters**: The function $$ P(t) $$ represents the population at time $$ t $$. The sine function oscillates between -1 and 1, which means the population will vary accordingly.

2. **Calculate the population at the start and end of the cycle**:
   - At $$ t = 0 $$:
     $$
     P(0) = 200 + 50 \sin(0) = 200 + 50 \cdot 0 = 200
     $$
   - At $$ t = 2\pi $$:
     $$
     P(2\pi) = 200 + 50 \sin(2\pi) = 200 + 50 \cdot 0 = 200
     $$

3. **Determine the maximum and minimum population during the cycle**:
   - The maximum population occurs when $$ \sin(t) = 1 $$:
     $$
     P_{\text{max}} = 200 + 50 \cdot 1 = 250
     $$
   - The minimum population occurs when $$ \sin(t) = -1 $$:
     $$
     P_{\text{min}} = 200 + 50 \cdot (-1) = 150
     $$

4. **Calculate the total population increase over one complete cycle**:
   - The total increase in population from the minimum to the maximum is:
     $$
     \text{Increase} = P_{\text{max}} - P_{\text{min}} = 250 - 150 = 100
     $$

Thus, the total population increase over one complete cycle (from $$ t=0 $$ to $$ t=2\pi $$) is $$ \boxed{100} $$.
The total population increase over one complete cycle is 100.

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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