The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at \( 12: 00 \) am and \( 12: 30 \mathrm{pm} \), with a depth of 2.5 m , while high tides occur at \( 6: 15 \) am and \( 6: 45 \mathrm{pm} \), with a depth of 5.5 m . Let \( t=0 \) be \( 12: 00 \mathrm{am} \). Write a cosine model, \( d=a \cos (b t)+k \), for the depth as a function of time. This amplitude is 1.5 \( a=9.5 \) \( \mathrm{X} \Rightarrow-1.5 \) The period is meters. \( b=\square \) hours.

To build your cosine model, let's first clarify the parameters. The amplitude \( a \) is the distance between the midline and the maximum or minimum depth. Given \( a = 1.5 \), that's confirmed. The height difference between high tide (5.5 m) and low tide (2.5 m) gives a total variation of \( 5.5 - 2.5 = 3 \, m \), with the average (midline depth) being \( k = \frac{5.5 + 2.5}{2} = 4 \, m \). Next, to find \( b \) which determines the period, note that one complete cycle of tide changes occurs from low tide at 12:00 am to the next low tide at 12:30 pm. This is a total of 12.5 hours for one complete cycle. The period \( T \) can be expressed as \( T = \frac{2\pi}{b} \). Therefore, to calculate \( b \): 1. Determine \( T \) in hours: \( T = 12.5 \). 2. Set \( T = \frac{2\pi}{b} \) leading to \( b = \frac{2\pi}{T} = \frac{2\pi}{12.5} = \frac{\pi}{6.25} \). Putting it all together, the cosine model is: \( d = 1.5 \cos\left(\frac{\pi}{6.25}t\right) + 4 \), where \( t \) is measured in hours. Now, don't forget to check your work for fun! Converting the period into a fraction gives you the ratio of the cycles—perfect for a group study session or a friendly math duel!