Problem
4. In a busy harbor, the time difference between successive high tides is about 12.3 hours. The water varies by 2.4 meters between high and low tide. The first high tide is 4.7 meters. Let $H \mathrm{~m}$ be the height of the tide $t$ hours after the first high tide. a. Explain why a cosine model (instead of a sine model) makes sense to use in this situation.
Solution
Step 1 ：Determine the amplitude, period, and vertical shift: amplitude = \(\frac{2.4}{2} = 1.2\), period = 12.3, vertical_shift = \(\frac{4.7 + 2.4}{2} = 3.5\)
Step 2 ：Write the cosine model: \(H(t) = 1.2 \cos\left(\frac{2\pi t}{12.3}\right) + 3.5\)
Step 3 ：\boxed{H(t) = 1.2 \cos\left(\frac{2\pi t}{12.3}\right) + 3.5}
