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Question

Coastal areas experience tides which is where the ocean periodically gets to high and low points. Tides can be modeled with a sinusoidal (sine or cosine) function. At one beach, the high tide is 4 feet above mean sea level and the low tide is 4 feet below sea level. The length of time between high and low tide is 4.5 hours.

If high tide is at time  hours, give the function H(t) that describes the height of the tide relative to sea level  hours after the high tide.

Answer 1

Coastal areas experience tides which is where the ocean periodically gets to high and low points. Tides can be modeled with a sinusoidal (sine or cosine) function. At one beach, the high tide is 4 feet above mean sea level and the low tide is 4 feet below sea level. The length of time between high and low tide is 4.5 hours.

If high tide is at time t=0 hours, give the function H(t) that describes the height of the tide relative to sea level t hours after the high tide.

1. Since high tide is given at time=0, this would be a cosine function (starts high, then dips; vs a sine function that starts at zero height and increasing).
2. h(t) = a cos(b(x-h)) + k; where: |a|=amplitude; period= (2π/|b|); h=horizontal shift, k vertical shift
3. High Tide = +4'; Low Tide = -4', so the amplitude is high to low/2: a = 8/2; a = 4
4. 1 period = 4.5(2) to complete a cycle = 9 hours
5. period = 2π/|b|=9 hours; |b|=(2/9)π
6. There is no vertical or horizontal shift (no h or k values here)
7. H(t) = 4 cos((2/9)π(t))

Does this help? If you need a visual, try entering the equation into Desmos graphing calculator on line.