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
