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 $$ t=2\pi $$).

Question

UpStudy AI · Accepted Answer

To determine the total population increase over one complete cycle of the function $$ P(t) = 200 + 50 \sin(t) $$ from $$ t=0 $$ to $$ t=2\pi $$, we will follow these steps:

1. **Evaluate the population at the start and end of the cycle**:
   - Calculate $$ P(0) $$.
   - Calculate $$ P(2\pi) $$.

2. **Find the total population increase**:
   - Subtract the initial population from the final population.

Let's start with the calculations.

### Step 1: Evaluate the population at $$ t=0 $$ and $$ t=2\pi $$

1. Calculate $$ P(0) $$:
   $$
   P(0) = 200 + 50 \sin(0) = 200 + 50 \cdot 0 = 200
   $$

2. Calculate $$ P(2\pi) $$:
   $$
   P(2\pi) = 200 + 50 \sin(2\pi) = 200 + 50 \cdot 0 = 200
   $$

### Step 2: Find the total population increase

Now, we find the total population increase over the cycle:
$$
\text{Total Increase} = P(2\pi) - P(0) = 200 - 200 = 0
$$

### Conclusion

The total population increase over one complete cycle from $$ t=0 $$ to $$ t=2\pi $$ is $$ 0 $$. This means that the population returns to its original value after one complete cycle.
The population increase over one complete cycle is 0.

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 ).

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions