Attendance at a state park throughout the year is found to be periodic and can be modeled by a sine function. The attendance ranges from a low of approximately $$ 1,000,000 $$ visitors in September to a high of approximately $$ 2,000,000 $$ visitors in March. If $$ t $$ is the month number, where $$ t=1 $$ is January, and $$ N(t) $$ is the attendance, in millions, of visitors, which of the functions can be used to model this behavior? $$ N(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+1.5 $$ $$ N(t)=2 \sin \left(\frac{\pi}{6} t\right)-1 $$ $$ N(t)=1.5 \sin \left(\frac{\pi}{6} t\right)+0.5 $$ $$ N(t)=0.5 \sin (2 \pi t)+1.5 $$

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UpStudy AI · Accepted Answer

To model the attendance at the state park using a sine function, we need to determine the appropriate amplitude, vertical shift, and period based on the given information.

**Given:**
- **Attendance Range:** From 1,000,000 (low in September) to 2,000,000 (high in March) visitors.
- **Month Representation:** $$ t = 1 $$ is January, $$ t = 2 $$ is February, ..., $$ t = 12 $$ is December.

**Step 1: Determine the Amplitude and Vertical Shift**

The attendance ranges from 1 to 2 million (in millions).

- **Amplitude (A):**
  $$
  A = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{2 - 1}{2} = 0.5
  $$
  
- **Vertical Shift (D):**
  $$
  D = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{2 + 1}{2} = 1.5
  $$

**Step 2: Determine the Period**

The period of the sine function should represent a full year (12 months).

- **Standard Period of Sine Function:** $$ 2\pi $$
- **Desired Period ($$ T $$):** 12 months
- **Angular Frequency ($$ \omega $$):**
  $$
  \omega = \frac{2\pi}{T} = \frac{2\pi}{12} = \frac{\pi}{6}
  $$

**Step 3: Construct the Sine Function**

Using the general form of the sine function:
$$
N(t) = A \sin(\omega t) + D
$$

Substituting the values:
$$
N(t) = 0.5 \sin\left(\frac{\pi}{6} t\right) + 1.5
$$

**Verification of Options:**

- **Option A:** $$ N(t) = 0.5 \sin\left(\frac{\pi}{6} t\right) + 1.5 $$  
  - **Amplitude:** 0.5  
  - **Vertical Shift:** 1.5  
  - **Period:** 12 months  
  - **✓ Correct**

- **Option D:** $$ N(t) = 0.5 \sin(2\pi t) + 1.5 $$  
  - **Period:** 1 month (incorrect)  
  - **✗ Incorrect**

- **Options B and C** do not match the required amplitude and vertical shift.

**Conclusion:**

The correct function that models the attendance behavior is:

$$
N(t) = 0.5 \sin\left(\frac{\pi}{6} t\right) + 1.5
$$

**Answer:**
$$
N(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+1.5
$$
The correct function to model the attendance is: $$ N(t) = 0.5 \sin\left(\frac{\pi}{6} t\right) + 1.5 $$

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