Tides The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Admini

Question

Tides The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, April 26, 2014, in Charleston, South Carolina, high tide occurred at 6:30 am (6.5 hours) and low tide occurred at 12:24 pm (12.4 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 5.86 feet, and the height of the water at low tide was − 0.38 foot.
(a) Approximately when will the next high tide occur?
(b) Find a sinusoidal function of the form
y = A sin(ωx – ϕ) + B
that models the data.
(c) Use the function found in part (b) to predict the height of the water at 3 pm on April 26, 2014.

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Xavia 3 years 2021-07-20T20:53:45+00:00 1 Answers 3 views 0

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  1. kiet
    0
    2021-07-20T20:55:06+00:00
    Reply

    Answer:

    (a) The next tide will occur at 6:55pm

    (b) y = 3.12 \cos(\frac{24\pi}{149}(x - 6.5)) + 2.74

    (c) The height is: 2.904ft

    Step-by-step explanation:

    Given

    T_1 = 12hr:25min — difference between high tides’

    Solving (a): The next time a high tide will occur

    From the question, we have that:

    High =6:30am — The time a high tide occur

    The next time it will occur is the sum of High and T1

    i.e.

    Next = High + T_1

    Next = 6:30am + 12hr : 25min

    Add the minutes

    Next = 6:55am + 12hr

    Add the hours

    Next = 6:55pm

    Solving (b): The sinusoidal function

    Given

    High\ Tide = 5.86

    Low\ Tide = -0.38

    T = 12hr:25min — difference between consecutive tides

    Shift = 6.5hr

    The sinusoidal function is represented as:

    y = A\cos(w(x - C)) + B

    Where

    A = Amplitude

    A = \frac{1}{2}(High\ Tide - Low\ Tide)

    A = \frac{1}{2}(5.86 - -0.38)

    A = \frac{1}{2}(6.24)

    A = 3.12

    B = Mean

    B = \frac{1}{2}(High\ Tide + Low\ Tide)

    B = \frac{1}{2}(5.86 - 0.38)

    B = \frac{1}{2}(5.48)

    B = 2.74

    w = Period

    w = \frac{2\pi}{T}

    w = \frac{2\pi}{12:25}

    Convert to hours

    w = \frac{2\pi}{12\frac{25}{60}}

    Simplify

    w = \frac{2\pi}{12\frac{5}{12}}

    As improper fraction

    w = \frac{2\pi}{\frac{149}{12}}

    Rewrite as:

    w = \frac{2\pi*12}{149}

    w = \frac{24\pi}{149}

    C = shift

    C=6.5

    So, we have:

    y = A\cos(w(x - C)) + B

    y = 3.12 \cos(\frac{24\pi}{149}(x - 6.5)) + 2.74

    Solving (c): The height at 3pm

    At 3pm, the value of x is:

    x=3:00pm - 6:30am

    x=9.5hrs

    So, we have:

    y = 3.12 \cos(\frac{24\pi}{149}(x - 6.5)) + 2.74

    y = 3.12 \cos(\frac{24\pi}{149}(9.5 - 6.5)) + 2.74

    y = 3.12 \cos(\frac{24\pi}{149}(3)) + 2.74

    y = 3.12 \cos(\frac{72\pi}{149}) + 2.74

    y = 3.12 *0.0527 + 2.74

    y = 2.904ft

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