Some of the highest tides in the world occur in the Bay of...

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about $2.0 \mathrm{~m}$ and at high tide it is about $12.0 \mathrm{~m}$. The natural period of oscillation is about 12 hours and on a particular day, high tide occurred at $6: 45$ AM. Find a function involving the cosine function that models the water depth $D(t)$ (in meters) as a function of time $t$ (in hours after midnight) on that day.

Key Concepts

Period and Angular Frequency

The period is the time required for one complete cycle of the tidal oscillation. Angular frequency, computed as 2? divided by the period, scales the cosine function so that it fits the observed cycle duration. This concept is essential for accurately modeling the timing of tides.

Sinusoidal Functions

Sinusoidal functions, including the cosine function, are used to model periodic behavior in natural phenomena. They are characterized by repeating patterns, making them ideal for describing cycles like tidal movements, where the water depth oscillates in a regular manner over time.

Amplitude

The amplitude represents the maximum deviation of the function from its midline. In the context of tidal modeling, it is calculated as half the difference between the high tide and low tide levels, indicating how much the water depth rises or falls relative to the average level.

Vertical Shift (Midline)

The vertical shift, or midline, is the baseline level around which the sinusoidal function oscillates. It is obtained by averaging the maximum and minimum values, providing a central value from which the amplitude is measured, which in tidal models represents the mean water depth.

Phase Shift

The phase shift is a horizontal translation of the cosine function that aligns the model with the observed data. It adjusts the function so that its maximum or minimum occurs at the correct time, ensuring that the model accurately reflects the timing of events like the high tide observed at a specific hour.

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