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Over a 24-hour period, the tide in a harbor can be modeled by one period of a
sinusoidal function. The tide measures 5 ft at midnight, rises to a high of 9.5 ft, falls
to a low of 0.5 ft, and then rises to 5 ft by the next midnight.

Give the value of each given that x represents time in hours since the beginning of
the 24 hour period that models the situation.

Amplitude =
Equation of midline
Y=

Answer :

The sinusoidal function that models the situation is:

[tex]y = 4.5\sin{\left(\frac{\pi}{12}\right)x} + 5[/tex]

From the function, we have that:

  • The amplitude is of 4.5 ft.
  • The mid-line equation is of y = 5.

What is a sinusoidal function?

A sinusoidal function is a trigonometric function, and has the following format, considering no phase shift:

[tex]y = A\sin{(Bx)} + C[/tex]

In which:

  • A is the amplitude, which is the twice the difference between the largest and smallest value.
  • The period is of [tex]\frac{2\pi}{B}[/tex].
  • C is the vertical shift, and the mid-line equation is [tex]y = C[/tex].

In this problem, the high is of 9.5 ft and the low is of 0.5 ft, hence, for the amplitude, we have that:

[tex]2A = 9[/tex]

[tex]A = \frac{9}{2}[/tex]

[tex]A = 4.5[/tex]

With an amplitude of 4.5, the standard function would vary between -4.5 ft and 4.5 ft, while in this problem the variation is from 0.5 ft to 9.5 ft, hence the vertical shift is of:

[tex]C = 9.5 - 4.5 = 4.5 - (-0.5) = 5[/tex]

Which means that the mid-line equation is of y = 5.

Period of 24 hours, hence:

[tex]\frac{2\pi}{B} = 24[/tex]

[tex]24B = 2\pi[/tex]

[tex]B = \frac{\pi}{12}[/tex]

Hence, the function is:

[tex]y = 4.5\sin{\left(\frac{\pi}{12}\right)}x + 5[/tex]

You can learn more about sinusoidal functions at https://brainly.com/question/26384970

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