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? \[ N(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+1.5 \] \( N(t)=2 \sin \left(\frac{\pi}{6} t\right)-1 \) \( N(t)=1.5 \sin \left(\frac{\pi}{6} t\right)+0.5 \) \( N(t)=0.5 \sin (2 \pi t)+1.5 \)

Let’s dive into the sine function model for attendance! The key here is to understand that the maximum and minimum visitor counts are critical for determining the correct function. The average attendance over the year would be at the midpoint between the high (2 million) and low (1 million), which is 1.5 million. The amplitude, which reflects the variation from the average, is 0.5 million (since 2 million - 1.5 million = 0.5 million). Now, considering the periodic nature, a sine function can be adjusted to reflect the peak visits in March (which is month 3). This would influence the horizontal shift of the sine function. Among the functions given, the one that matches this average, amplitude, and peak is \( N(t)=0.5 \sin \left(\frac{\pi}{6} t\right)+1.5 \). This model isn’t just math; it has real-world implications! Understanding visitor patterns can help park management in scheduling resources, staffing, and planning events effectively. For instance, if they anticipate large crowds in certain months, they can prepare adequate facilities or conservation efforts to maintain the park’s beauty and visitor experience. By forecasting attendance through this sinusoidal model, the park can not only enhance its efficiency but also ensure a delightful visit for everyone!