The height, \( h \), in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. Which of the following equations can be used to model the height as a function of time, \( t \), in hours? Assume that the time at \( t=0 \) is \( 12: 00 \) a.m. \( n=0.5 \cos \left(\frac{\pi}{12} t\right)+9.5 \) \( n=0.5 \cos \left(\frac{\pi}{6} t\right)+9.5 \) \( n=\cos \left(\frac{\pi}{12} t\right)+9 \) \( n=\cos \left(\frac{\pi}{6} t\right)+9 \)

To model the height of the tip of the hour hand of a wall clock as a function of time, we need to consider the following: 1. The height varies between 9 feet and 10 feet, which means the average height is \( \frac{9 + 10}{2} = 9.5 \) feet. 2. The amplitude of the variation is \( \frac{10 - 9}{2} = 0.5 \) feet. 3. The hour hand completes one full rotation (from 12:00 back to 12:00) in 12 hours. Given this information, we can model the height \( h(t) \) as a cosine function, which starts at its maximum value when \( t = 0 \) (at 12:00 a.m.). The general form of the cosine function is: \[ h(t) = A \cos(Bt) + C \] Where: - \( A \) is the amplitude, - \( B \) is related to the period of the function, - \( C \) is the vertical shift (average height). From our analysis: - The amplitude \( A = 0.5 \), - The average height \( C = 9.5 \). Next, we need to determine \( B \). The period of the cosine function is given by \( \frac{2\pi}{B} \). Since the hour hand completes one full rotation in 12 hours, the period is 12 hours: \[ \frac{2\pi}{B} = 12 \implies B = \frac{2\pi}{12} = \frac{\pi}{6} \] Now we can write the function: \[ h(t) = 0.5 \cos\left(\frac{\pi}{6} t\right) + 9.5 \] This matches the form of the first option provided: \[ n = 0.5 \cos\left(\frac{\pi}{6} t\right) + 9.5 \] Thus, the correct equation to model the height of the tip of the hour hand as a function of time \( t \) in hours is: \[ \boxed{n = 0.5 \cos\left(\frac{\pi}{6} t\right) + 9.5} \]