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 $$ 12: 30 $$ pm, with a depth of 2.5 m , while high tides occur at $$ 6: 15 \mathrm{am} $$ and $$ 6: 45 \mathrm{pm} $$, with a depth of 5.5 m . Let $$ t=0 $$ be $$ 12: 00 \mathrm{am} $$. Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.

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A **cosine function** would serve as a simpler model for the tidal depth in this scenario. Here's why:

1. **Starting Point (t = 0):**
   - At $$ t = 0 $$ (12:00 am), the depth is at a **minimum** (2.5 m).
   - The standard cosine function, $$ \cos(0) $$, starts at its **maximum** value. However, by using a phase shift or simply negating the cosine function, you can align the minimum at $$ t = 0 $$.
   - For example, $$ D(t) = 4 - 1.5 \cos\left(\frac{2\pi}{12.5}t\right) $$ starts at the minimum depth when $$ t = 0 $$.

2. **Symmetry of Tides:**
   - High tides and low tides alternate in a regular, symmetric pattern.
   - The cosine function naturally models such symmetric oscillations, making it easier to fit both high and low tides without introducing additional shifts or complexities.

3. **Ease of Phase Alignment:**
   - Aligning the cosine function to start at a minimum (or maximum) typically requires simpler adjustments compared to the sine function, which might necessitate more complex phase shifts to achieve the same alignment.

In summary, the cosine function's inherent properties make it more straightforward to model the periodic rise and fall of tidal depths, especially when starting at an extreme value like a minimum.

**Answer:**  
A cosine function is more straightforward because the depth begins at a minimum, making a cosine model simpler for this periodic behavior.
A cosine function is simpler for modeling the tidal depth because it naturally starts at the minimum depth at $$ t = 0 $$, aligning well with the given data.

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