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. \( h=0.5 \cos \left(\frac{\pi}{12} t\right)+9.5 \) \( h=0.5 \cos \left(\frac{\pi}{6} t\right)+9.5 \) \( h=\cos \left(\frac{\pi}{12} t\right)+9 \) \( h=\cos \left(\frac{\pi}{6} t\right)+9 \)

To model the height of the tip of the hour hand of a wall clock as a function of time \( t \), it's important to know the period of the function and its vertical shift. The hour hand completes a full rotation in 12 hours, so the equation should have a factor of \(\frac{\pi}{12}\) to reflect this period. The height varies between 9 feet and 10 feet, averaging to 9.5 feet. The amplitude, half the distance between maximum and minimum, is 0.5 feet. Thus, the correct equation is \( h = 0.5 \cos \left(\frac{\pi}{12} t\right) + 9.5 \). Now, let's sprinkle in some fun facts! The clock we refer to today has roots in ancient civilizations; the Egyptians created sundials over 3,500 years ago! They laid the groundwork for our modern understanding of timekeeping, showing just how integral time has been to human progress and structure. Have you ever tried timing something with a clock? A common mistake is not considering the time unit accurately. For instance, if you're observing the hour hand movements, remember that it moves steadily, not in jumps, which can be confusing at first! Stay focused on fractions of hours to accurately track the hour hand's height.