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To determine which trigonometric function best models the population of grasshoppers, let's analyze the graph and compare it to the given options.

### Step-by-Step Analysis:

1. **Determine the Amplitude:**
   - The graph shows that the population fluctuates between 3000 and 11000.
   - The amplitude of a trigonometric function \(P(t) = A \cos(Bt) + C\) is given by the distance from the midline to the peak or trough.  
   - Here, the midline (average of the maximum and minimum values) is:
     \[
     \text{Midline} = \frac{3000 + 11000}{2} = 7000.
     \]
   - The amplitude is:
     \[
     \text{Amplitude} = 11000 - 7000 = 4000.
     \]

2. **Determine the Midline:**
   - As calculated above, the midline is 7000. This is the value that the function oscillates around.

3. **Determine the Period:**
   - A full cycle occurs from week 0 to week 6.
   - The period \(T\) for a trigonometric function \(P(t) = A \cos(Bt) + C\) is given by \(T = \frac{2\pi}{B}\).
   - Here, the period is 6 weeks. So:
     \[
     6 = \frac{2\pi}{B} \implies B = \frac{2\pi}{6} = \frac{\pi}{3}.
     \]

4. **Analyze the Options:**
   - The correct function should have:
     - Amplitude = 4000
     - Midline = 7000
     - Period of \(6\) weeks, which corresponds to \(B = \frac{\pi}{3}\).

   - **Option A:** \(P(t) = 7000 + 4000 \sin\left(\frac{2\pi}{3}t\right)\)  
     - This has the correct midline and amplitude but the period would be 3 weeks (not correct).

   - **Option B:** \(P(t) = 7000 + 4000 \cos\left(\frac{2\pi}{3}t\right)\)  
     - This has the correct midline, amplitude, and frequency. 

   - **Option C:** \(P(t) = 5000 + 2000 \sin\left(\frac{2\pi}{3}t\right)\)  
     - The midline is 5000, which is incorrect.

   - **Option D:** \(P(t) = 5000 + 2000 \cos\left(\frac{2\pi}{3}t\right)\)  
     - The midline is 5000, which is incorrect.

### Conclusion:
The correct answer is **Option B:** \(P(t) = 7000 + 4000 \cos\left(\frac{\pi}{3} t\right)\).

Would you like more details on this solution, or do you have any questions?

### Related Questions:
1. How do you find the period of a trigonometric function?
2. What is the difference between a sine and cosine function in terms of their graphs?
3. How can the amplitude of a trigonometric function affect the shape of the graph?
4. What are other possible applications of trigonometric models in real-world scenarios?
5. How would you determine the phase shift of a trigonometric function?

### Tip:
When analyzing a trigonometric function, always check the amplitude, period, and midline to ensure that the model matches the data accurately.

The population of grasshoppers in a natural habitat was observed for 9 weeks. A trigonometric model of the data was made. The graph shows the population of grasshoppers, P(t), after t weeks from the beginning of the study. Which trigonometric function models the population of grasshoppers?

Analyze a graph showing the population of grasshoppers in a natural habitat over 9 weeks and determine which trigonometric function best models the population. This problem involves concepts such as amplitude, period, and the midline of trigonometric functions. Suitable for high school students.

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