At a dock on the east coast, low tide occurs at 3 pm wi th a water depth of 5 ft. The depth at high tide is 83 ft. High tide occurs every 6 hours. Explain how to find the sinusoidal function that models the depth in terms of time, x.

The sinusoidal function that model the depth in terms of time, 'x', is presented here as follows;

f(x) = 39·sin((π/3)·x - π) + 5

Step-by-step explanation:

A sinusoidal function is given by the general function as follows;

y = A·sin(B·x + C) + D

Where;

A = The amplitude = ( - )/2 = (The high tide - The low tide)/2 = (83 ft. - 5 ft.)/2 = 39 ft.

The period, T = 2·π/b = 6 hours

∴ B = 2·π/T = 2·π/6 = π/3

D = The vertical shift = The low tide = 5 ft.

The horizontal phase shift, 'C', is given as follows;

3 hrs = -C/B = -C/(π/(3 hr))

C = 3 hrs × -(π/(3 hr)) = -π

∴ C = -π

y = 39·sin((π/3)·x - π) + 5

The sinusoidal function that model the depth in terms of time, 'x', is therefore given as follows;

f(x) = 39·sin((π/3)·x - π) + 5