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 \( 12: 00 \mathrm{a} \).m. \( n=0.5 \cos \left(\frac{\pi}{12} t\right)+9.5 \) \( n=0.5 \cos \left(\frac{\pi}{6} t\right)+9.5 \) \( n=\cos \left(\frac{\pi}{12} t\right)+9 \) \( n=\cos \left(\frac{\pi}{6} t\right)+9 \)

The height of the tip of the hour hand of a clock can be modeled using a cosine function because the motion of the hour hand is periodic. It completes one full revolution (back to its starting position) every 12 hours, and thus, the period of the cosine function will be \( \frac{\pi}{6} \) radians per hour. Now let's look at the height range. The hand varies from 9 feet to 10 feet, giving us a midline of \( 9.5 \) feet, and an amplitude of \( 0.5 \) feet. So, the correct model that fits this description is \( n = 0.5 \cos \left(\frac{\pi}{6} t\right) + 9.5 \). Now, isn't it interesting how clocks have transitioned through history? The oldest known clocks were water clocks used in ancient Egypt around 1400 BC, which measured time with the steady flow of water! These ancient devices helped civilizations keep track of time for agricultural and religious purposes, paving the way for the intricate timekeeping we have today. For those looking to deepen their understanding of periodic functions, a fascinating journey awaits in exploring the world of harmonic motion! Delve into topics like sine and cosine functions, and their applications in real-world scenarios such as sound waves, or even in analyzing musical tones. Books on calculus and physics would provide a treasure trove of knowledge on these cyclic behaviors!