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Iver a 24-hour period, the tide in a harbor can be modeled by one period of a sinusoidal function. The tide measures 4.35 ft at midnight, rises to a high of 8.3 ft, falls to a low of 0.4 ft, and then rises to 4.35 ft by the next midnight.

What is the equation for the sine function f(x), where x represents time in hours since the beginning of the 24-hour period, that models the situation?​

Respuesta :

Answer:

[tex]f(x)=3.95sin(\frac{\pi}{12}x)+4.35[/tex]

Step-by-step explanation:

Recall:

  • Sinusoidal Function -> [tex]f(x)=a*sin(bx+c)+d[/tex]
  • Amplitude -> [tex]a[/tex]
  • Period -> [tex]\frac{2\pi}{|b|}[/tex]
  • Phase Shift -> [tex]-\frac{c}{b}[/tex]
  • Vertical Shift/Midline -> [tex]d[/tex]

Given:

  • Amplitude -> [tex]a=8.3-4.35=3.95[/tex]
  • Period -> [tex]\frac{2\pi}{|b|}=\frac{2\pi}{|24|}=\frac{2\pi}{24}=\frac{\pi}{12}[/tex]
  • Phase Shift -> [tex]-\frac{c}{b}=-\frac{0}{24}=0[/tex]
  • Vertical Shift/Midline -> [tex]d=\frac{8.3+0.4}{2}=\frac{8.7}{2}=4.35[/tex]

Conclusion:

The equation that models the situation is [tex]f(x)=3.95sin(\frac{\pi}{12}x)+4.35[/tex]

Hope this helped! I've attached a graph of the function so you can understand it better!

Ver imagen goddessboi

Modelling a situation can be done using some mathematical constructs if possible. The sin function modelling this tidal situation is [tex]f(x) = 3.95\sin(\dfrac{\pi x}{12}) + 4.35[/tex]

How does sine function works?

Suppose that we've got

[tex]f(x) = a\sin(bx + c) + d[/tex]

It has got

  • Amplitude or maximum height from average motion horizontal axis  = a

And thus, the minimum low from horizontal axis is -a (sin ranges from -1 to 1, and multiplying a to it make it range from -a to a).

  • Period of wave: [tex]\dfrac{2\pi}{b}[/tex]
  • Phase shift (horizontal left shift) = c/b
  • Horizontal line around which wave moves is: y = d

For the given case, the maximum high is 8.3, whereas normal level is 4.35, which means it rises 8.3-4.35 = 3.95 units high from normal level.

Thus, a = 3.95

Since the average value axis is 4.35, thus we have d = 4.35

SInce the motion starts from 00:00 (midnight), thus, phase shift is 0, or c/b = 0, or c = 0

It comes from 4.35 to 4.35 in 24 hours, thus, its period is 24 hours

or

[tex]24 = \dfrac{2\pi}{b}\\\\b = \dfrac{\pi}{12}[/tex]

Thus, putting all values, we get the sin function modelling this tidal situation as:

[tex]f(x) = a\sin(bx + c) + d\\\\f(x) = 3.95\sin(\dfrac{\pi x}{12}) + 4.35[/tex]

Thus,

The sin function modelling this tidal situation is

[tex]f(x) = 3.95\sin(\dfrac{\pi x}{12}) + 4.35[/tex]

Learn more about sin function here:

https://brainly.com/question/13927126

Ver imagen astha8579

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