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The water level varies from 12 inches at low tide to 52 inches at high tide. Low tide occurs at 9:15 a.m. and high tide occurs at 3:00 p.m. What is a cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m.?

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Answer:

Answer

Explanation:

To model the variation in inches above and below the water level as a function of time in hours, we can use a cosine function.

First, we need to determine the amplitude and period of the function. The amplitude is half the difference between the high and low tides, which is (52 - 12)/2 = 20 inches. The period is the time it takes for one complete cycle, which is the time between consecutive high tides or low tides. In this case, the period is 6 hours, since high tide occurs every 6 hours.

The general form of a cosine function is:

y = A cos (Bx - C) + D

where A is the amplitude, B is the frequency (in radians per unit time), C is the phase shift (in radians), and D is the vertical shift.

Substituting the values we found, we get:

y = 20 cos ((2π/6)(x - 1.75)) + 32

where x is the time in hours since 9:15 a.m., and 1.75 is the phase shift (since high tide occurs at 3:00 p.m., which is 5 hours and 45 minutes after 9:15 a.m.). The vertical shift D is 32 inches, which is the average of the low and high tides.

Therefore, the cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m. is:

y = 20 cos ((2π/6)(x - 1.75)) + 32

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