Modeling Periodic Phenomena with Trigonometric Functions

Question

Throughout any given month, the maximum and minimum ocean tides follow a periodic pattern. Last year, at a certain location on the California coast, researchers recorded the height of low tide, with respect to sea level, each day during the month of July. The lowest low tide was first measured on July 11, at -1.4 feet. The highest low tide was first measured on July 4, at 1.8 feet. The average low tide for the month of July was measured to be 0.2 feet.

Part A

Which curve would you choose to model this function, sine or cosine? Give your reasons.

Part B

Find the amplitude of the function. Explain its meaning in the context.

Part E

Find the period of the function. Explain its meaning in the context.

Part C

Find the vertical shift of the function. Explain its meaning in the context.

Part D

Find the phase shift of the function. Explain its meaning in context.

Part E

Write the function for the curve from all of the key features you just found.

Part F

What are the days of the month when the low tide is projected to be the average height? What do you suppose a decimal value for a day in the month means?

Part G

The actual low tide height recorded on a given day could vary from the function created as a model. Give some reasons for why you think this happens.

Answer 1

A: Sine and cosine are the same curve, just shifted on the x-axis. This model is closer to a cosine, because the value at the beginning of the month is close to its highest value, rather than being closer to a sine (which would have the starting value be zero).

B: The amplitude of a sine/cosine function is half the difference between the highest and lowest values (so an amplitude 1 sine/cosine has a difference of 2 between its highest and lowest values). The amplitude here is 1.6 feet.

E: It takes the tide 7 days to go from its highest low to its lowest low, so the period is twice that, or 14 days.

C: The average value for a pure sine/cosine is zero. This model is shifted up by 0.2 feet (the average low tide).

D: The phase shift would be zero if the highest low was on the first day. It's on the 4th day, so the phase shift is 4 days later in the month.

E: Using radians, with D the day of the month: 1.6 cos(2 π (D - 4)/ 14) + 0.2

F: The tide should be at its average height on days 7.5, 14.5, 21.5, etc. As there are two low tides in each day, I would guess this means the second one in the day, assuming the measurements mentioned in the problem were for the first low tides of the day.

G: We're using a cosine function to model the tides, but the height of the low tide actually depends on the positions of the sun and moon relative to the earth. The cosine is only an approximation. Also, what happens to the ocean shore on a given day may depend on the weather.