In​ predator-prey relationships, the populations of the predator and prey are often cyclical. In a conservation​ area, rangers monitor the population of carnivorous animals and have determined that the population can be modeled by the function Upper P left parenthesis t right parenthesis equals 40 cosine left parenthesis StartFraction pi t Over 6 EndFraction right parenthesis plus 110
where t is the number of months from the time monitoring began. Use the model to estimate the population of carnivorous animals in the conservation area after 10 ​months, 28 ​months, and 42 months.

The population of carnivorous animals in the conservation area is modeled by the function:

\[
P(t) = 40 \cos\left(\frac{\pi t}{6}\right) + 110
\]

where \(t\) is the number of months from the time monitoring began. To estimate the population at specific times, we'll plug in the values of \(t\) (10 months, 28 months, and 42 months) into the function \(P(t)\).

### Calculations:

1. **For \(t = 10\) months:**

\[
P(10) = 40 \cos\left(\frac{\pi \times 10}{6}\right) + 110
\]

Simplify the fraction inside the cosine:

\[
P(10) = 40 \cos\left(\frac{10\pi}{6}\right) + 110 = 40 \cos\left(\frac{5\pi}{3}\right) + 110
\]

The value of \(\cos\left(\frac{5\pi}{3}\right)\) is \(\frac{1}{2}\), so:

\[
P(10) = 40 \times \frac{1}{2} + 110 = 20 + 110 = 130
\]

2. **For \(t = 28\) months:**

\[
P(28) = 40 \cos\left(\frac{\pi \times 28}{6}\right) + 110
\]

Simplify the fraction inside the cosine:

\[
P(28) = 40 \cos\left(\frac{28\pi}{6}\right) = 40 \cos\left(\frac{14\pi}{3}\right)
\]

\(\frac{14\pi}{3}\) is more than \(2\pi\), so we subtract \(2\pi\) (which is \(6\pi/3\)) twice to bring it within the range of \([0, 2\pi]\):

\[
\frac{14\pi}{3} - 2 \cdot \frac{6\pi}{3} = \frac{14\pi}{3} - \frac{12\pi}{3} = \frac{2\pi}{3}
\]

The value of \(\cos\left(\frac{2\pi}{3}\right)\) is \(-\frac{1}{2}\), so:

\[
P(28) = 40 \times \left(-\frac{1}{2}\right) + 110 = -20 + 110 = 90
\]

3. **For \(t = 42\) months:**

\[
P(42) = 40 \cos\left(\frac{\pi \times 42}{6}\right) + 110
\]

Simplify the fraction inside the cosine:

\[
P(42) = 40 \cos\left(\frac{42\pi}{6}\right) = 40 \cos\left(7\pi\right)
\]

Since \(7\pi\) is an odd multiple of \(\pi\), and \(\cos(\pi) = -1\):

\[
P(42) = 40 \times (-1) + 110 = -40 + 110 = 70
\]

### Results:

- The population of carnivorous animals after **10 months** is **130**.
- The population after **28 months** is **90**.
- The population after **42 months** is **70**.

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. How does the cosine function affect the periodicity of the population model?
2. What is the significance of the amplitude in the population function?
3. How would changing the coefficient of \(t\) inside the cosine function affect the population model?
4. What would the population be after 60 months?
5. Can we use this model to predict the population indefinitely? Why or why not?

**Tip:** The cosine function has a period of \(2\pi\). When calculating, always bring the angle back into the range \([0, 2\pi]\) for easier cosine evaluation.

Explore how to estimate carnivorous animal populations in a conservation area using a cosine function model. Learn about periodic functions and trigonometry applied to population dynamics. Includes calculations for 10 months, 28 months, and 42 months.

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