Sagot :
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]