The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let [tex]t=0[/tex] be 12:00 am. Write a cosine model, [tex]d=a \cos (b t)+k[/tex], for the depth as a function of time.
This amplitude is [tex]1.5[/tex] meters.
The period is hours.
[tex]b=[/tex]
Answer (2)
The period of the tide cycle is the time between two consecutive low tides, which is 12.5 hours.
Calculate b using the formula b = p er i o d 2 π .
Substitute the period into the formula: b = 12.5 2 π ≈ 0.50265 .
The period is 12.5 hours and b = 0.50265 .
Explanation
Understanding the Problem We are given that the depth of the water at the end of a pier changes periodically with the tides. Low tides occur at 12:00 am ( t = 0 ) and 12:30 pm, with a depth of 2.5 m. High tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. We want to write a cosine model, d = a cos ( b t ) + k , for the depth as a function of time. We are given that the amplitude is 1.5 m and a = − 1.5 . We need to find the period and the value of b .
Calculating the Period The time between two consecutive low tides is the period of the tide cycle. The low tides occur at 12:00 am and 12:30 pm. Therefore, the period is 12 hours and 30 minutes, which is 12.5 hours.
Finding b We know that the period is related to b by the formula: p er i o d = b 2 π . We can rearrange this to solve for b : b = p er i o d 2 π .
Calculating b Substituting the value of the period (12.5 hours) into the formula, we get: b = 12.5 2 π ≈ 12.5 2 × 3.14159 ≈ 0.50265 Therefore, b ≈ 0.50265 .
Final Answer The period of the tide cycle is 12.5 hours, and the value of b is approximately 0.50265.
Examples
Understanding tidal patterns is crucial for coastal navigation and marine activities. For instance, knowing the periodic changes in water depth helps ships safely navigate harbors, plan fishing activities, and manage coastal resources effectively. The cosine model we derived can predict water depths at different times, aiding in these practical applications and ensuring safer maritime operations.
The tidal model's period is 12.5 hours, leading to a value of b at approximately 0.50265. The complete cosine model for the water depth is d = 1.5 cos(0.50265t) + 4.0.
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