To determine which function best models the attendance at the state park, let's break down the given information:
1. Amplitude and Midline:
- The attendance varies from a low of [tex]\(1,000,000\)[/tex] visitors to a high of [tex]\(2,000,000\)[/tex] visitors.
- The midline (average value of the attendance) is the average of the high and low attendance:
[tex]\[
\text{Midline} = \frac{\text{High} + \text{Low}}{2} = \frac{2,000,000 + 1,000,000}{2} = 1,500,000
\][/tex]
- The amplitude (difference between the midline and either the high or low attendance) is:
[tex]\[
\text{Amplitude} = \text{High} - \text{Midline} = 2,000,000 - 1,500,000 = 500,000
\][/tex]
Hence, the amplitude is [tex]\(0.5\)[/tex] (since [tex]\(500,000\)[/tex] in millions is [tex]\(0.5\)[/tex]).
2. Period and Frequency:
- The period of the function must account for the yearly variation. Since the attendance has a full cycle within a year (12 months), the period is [tex]\(12\)[/tex].
- Generally, a sine function is represented by [tex]\(N(t) = A \sin(Bt) + D\)[/tex], where [tex]\(B\)[/tex] determines the period. The period [tex]\(P\)[/tex] is given by:
[tex]\[
P = \frac{2\pi}{B} = 12 \implies B = \frac{2\pi}{12} = \frac{\pi}{6}
\][/tex]
3. Phase Shift and Vertical Shift:
- The function reaches its maximum in March (month 3). The standard sine function [tex]\(\sin(x)\)[/tex] reaches its maximum at [tex]\(x = \frac{\pi}{2}\)[/tex]. To adjust the function such that its maximum occurs at [tex]\(t = 3\)[/tex], we need to account for a phase shift.
- The general form would then be [tex]\(N(t) = A \sin\left( B(t - \text{phase shift}) \right) + D\)[/tex].
4. Putting it all together:
- With the amplitude [tex]\(0.5\)[/tex], period modifier [tex]\(\frac{\pi}{6}\)[/tex], midline [tex]\(1.5\)[/tex] (vertical shift), and the above conditions, you can identify the correct function.
Given the options, our periodic model should be:
[tex]\[N(t) = 0.5 \sin \left( \frac{\pi}{6} t \right) + 1.5\][/tex]
Correct Function:
[tex]\[
\boxed{N(t) = 0.5 \sin \left( \frac{\pi}{6} t \right) + 1.5}
\][/tex]
