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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.

Sagot :

MrRoyalidn

Answer:

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

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

(c) The height is: 2.904ft

Step-by-step explanation:

Given

[tex]T_1 = 12hr:25min[/tex] --- difference between high tides'

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

From the question, we have that:

[tex]High =6:30am[/tex] --- The time a high tide occur

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

i.e.

[tex]Next = High + T_1[/tex]

[tex]Next = 6:30am + 12hr : 25min[/tex]

Add the minutes

[tex]Next = 6:55am + 12hr[/tex]

Add the hours

[tex]Next = 6:55pm[/tex]

Solving (b): The sinusoidal function

Given

[tex]High\ Tide = 5.86[/tex]

[tex]Low\ Tide = -0.38[/tex]

[tex]T = 12hr:25min[/tex] -- difference between consecutive tides

[tex]Shift = 6.5hr[/tex]

The sinusoidal function is represented as:

[tex]y = A\cos(w(x - C)) + B[/tex]

Where

[tex]A = Amplitude[/tex]

[tex]A = \frac{1}{2}(High\ Tide - Low\ Tide)[/tex]

[tex]A = \frac{1}{2}(5.86 - -0.38)[/tex]

[tex]A = \frac{1}{2}(6.24)[/tex]

[tex]A = 3.12[/tex]

[tex]B = Mean[/tex]

[tex]B = \frac{1}{2}(High\ Tide + Low\ Tide)[/tex]

[tex]B = \frac{1}{2}(5.86 - 0.38)[/tex]

[tex]B = \frac{1}{2}(5.48)[/tex]

[tex]B = 2.74[/tex]

[tex]w = Period[/tex]

[tex]w = \frac{2\pi}{T}[/tex]

[tex]w = \frac{2\pi}{12:25}[/tex]

Convert to hours

[tex]w = \frac{2\pi}{12\frac{25}{60}}[/tex]

Simplify

[tex]w = \frac{2\pi}{12\frac{5}{12}}[/tex]

As improper fraction

[tex]w = \frac{2\pi}{\frac{149}{12}}[/tex]

Rewrite as:

[tex]w = \frac{2\pi*12}{149}[/tex]

[tex]w = \frac{24\pi}{149}[/tex]

[tex]C = shift[/tex]

[tex]C=6.5[/tex]

So, we have:

[tex]y = A\cos(w(x - C)) + B[/tex]

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

Solving (c): The height at 3pm

At 3pm, the value of x is:

[tex]x=3:00pm - 6:30am[/tex]

[tex]x=9.5hrs[/tex]

So, we have:

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

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

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

[tex]y = 3.12 \cos(\frac{72\pi}{149}) + 2.74[/tex]

[tex]y = 3.12 *0.0527 + 2.74[/tex]

[tex]y = 2.904ft[/tex]

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