To model the change in depth over the next 24 hours, we can use a cosine function to represent the tide's periodic nature. Given that the water level is at average sea level at 2:00 am and that the tide is coming in, we can set up a cosine function to reflect these conditions.

Step-by-Step Solution:

1. Identify the Period of the Tide Cycle:

   • The tide typically completes one cycle (from high tide to low tide and back to high tide) over a period of about 12 hours.
   • Therefore, over a 24-hour period, we will see two full cycles of the tide.

2. Cosine Function for Tidal Changes:

   • A cosine function is commonly used to model periodic behavior, with the general form: $D(t) = A \cos\left(\frac{2\pi}{T} t - \phi\right) + C$ where:
     • $D(t)$ is the depth of water at time $t$,
     • $A$ is the amplitude (maximum deviation from the average depth),
     • $T$ is the period of the cycle,
     • $\phi$ is the phase shift, and
     • $C$ is the vertical shift (average sea level).

3. Set Constants Based on Information:

   • Amplitude ($A$): This value represents the maximum deviation from average sea level. Let’s assume that high tide raises the depth by $h$ meters above average, so $A = h$.
   • Period ($T$): Given that one full tide cycle takes about 12 hours, we set $T = 12$.
   • Vertical Shift ($C$): Since the average sea level is the midline of our function, we set $C = 0$.
   • Phase Shift ($\phi$): Since we are given that the depth is at average sea level at 2:00 am, we need to shift the cosine curve to ensure that the cycle aligns with this information.

4. Phase Shift Calculation ($\phi$):

   • We want the function to equal zero at $t = 2$ (i.e., at 2:00 am). Since the tide is coming in, the cosine function should be increasing at this point, meaning that $t = 2$ is halfway between low tide and high tide.
   • If we assume high tide occurs at around $t = 8$ (6 hours after average sea level at 2:00 am), then we can calculate the phase shift to ensure that $D(2) = 0$.

5. Construct the Equation:

   • The final function will be: $D(t) = h \cos\left(\frac{\pi}{6} (t - 2)\right)$ where:
     • $\frac{\pi}{6}$ comes from $\frac{2\pi}{12}$,
     • The shift $(t - 2)$ adjusts the function to match the average sea level at 2:00 am.

Final Equation:

$D(t) = h \cos\left(\frac{\pi}{6} (t - 2)\right)$

This equation models the depth $D(t)$ over the next 24 hours, where $t$ is the time in hours from midnight.

Let me know if you’d like further details on this or if you have any questions!

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Related Questions

1. How would the equation change if low tide occurred at 2:00 am instead?
2. What would the equation look like if the period of the tide cycle were 14 hours instead of 12?
3. How would adding a vertical shift to the equation affect the graph?
4. How can we adjust the equation if high tide happens at a different amplitude?
5. How does a phase shift impact the time when the tide reaches its peak?

Tip:

In trigonometric models, phase shifts help align cycles with specific times, making them essential for real-world applications like tidal patterns.