Final answer:
The water depth can be modeled using a sinusoidal equation of the form y(t) = A sin(Bt + C) + D. Given the maximum depth of 18 m at 3:00 am and the minimum depth of 4 m at 9:30 am, the amplitude, angular frequency, phase shift, and vertical shift can be determined. The resulting sinusoidal equation for the water depth is y(t) = 7 sin(0.97t) + 11.
Step-by-step explanation:
The water depth can be modeled using a sinusoidal equation of the form:
y(t) = A sin(Bt + C) + D
where:
A is the amplitude of the wave, which is half the difference between the maximum and minimum depths.
B is the angular frequency of the wave, which is 2π divided by the period of the wave.
C is the phase shift of the wave.
D is the vertical shift of the wave.
Given that there is a maximum depth of 18 m at 3:00 am and a minimum depth of 4 m at 9:30 am, we can determine the values of A, B, C, and D.
First, we find the amplitude:
A = (18 - 4)/2 = 7
Next, we find the period of the wave, which is the time it takes for one complete cycle:
Period = 9:30 am - 3:00 am = 6.5 hours
Using the formula for angular frequency, B = 2π/Period, we can calculate B:
B = 2π/(6.5) = 0.97 radians/hour
The phase shift, C, is determined by the time at which the wave reaches its maximum. Since the maximum depth is at 3:00 am, the wave is already at its peak at that time, so C = 0.
The vertical shift, D, is determined by the average of the maximum and minimum depths:
D = (18 + 4)/2 = 11 m
Therefore, the sinusoidal equation for the water depth is:
y(t) = 7 sin(0.97t) + 11