In one area of the Bay of Fundy, the tides cause the water level to rise to 7.2 m above average sea level and to drop to 7.2 m below average sea level. The time between the peak and the trough is approximately 6 hours. Determine the equation of a sinusoidal function that would model this situation assuming that at t=0 the trough of the tide is 2m.

the amplitude is clearly 7.2, so start with

y = 7.2 cos(kt)

The period is 2*6=12 hours, so k = 2π/12 = π/6

y = 7.2 sin(π/6 t)

Now we are stuck. The tide starts at 2m, but is that on the way down, or the way up? And how can that be the trough? The trough is at -7.2m.

Anyway, assuming you just mean that y(0) = 2, let's assume that the tide is rising then. That means
sin(π/6 t) = 2/7.2, so t=0.28

Shifting the graph to the left, that gives

y = 7.2 sin(π/6 t + 0.28)

http://www.wolframalpha.com/input/?i=7.2+sin(%CF%80%2F6+t%2B0.28)+for+-1%3Ct%3C12

To determine the equation of a sinusoidal function that models this situation, we can note the following key information:

Amplitude (A): The amplitude of a sinusoidal function is the distance from the midpoint to the peak (or trough) of the wave. In this case, the amplitude is 7.2 meters.

Midline (B): The midline represents the average sea level. In this case, the midline is the height at which the water level oscillates around, which is 0 meters.

Period (P): The period of a sinusoidal function is the time it takes for one complete cycle. In this case, the time between the peak and the trough is given as 6 hours. Since one complete cycle includes both the rise and the drop, the period is 6 hours * 2 = 12 hours.

Vertical Shift (C): The vertical shift represents any vertical displacement. At t=0, the trough of the tide is given as 2 meters below average sea level, which means there is a vertical shift of -2 meters.

Based on this information, we can use the general form of a sinusoidal function:

f(t) = A * sin((2π/P) * (t - phase shift)) + C

Substituting the known values:

A = 7.2
P = 12
C = -2

The phase shift is the horizontal displacement of the graph. In this case, the tide reaches the trough at t=0, so the phase shift is 0.

Substituting these values into the equation, we get:

f(t) = 7.2 * sin((2π/12) * (t - 0)) - 2

Simplifying further:

f(t) = 7.2 * sin((π/6) * t) - 2

Therefore, the equation of the sinusoidal function that models this situation is f(t) = 7.2 * sin((π/6) * t) - 2.

To determine the equation of a sinusoidal function that models the given tide situation, we need to identify the key elements of a sinusoidal function and use them to form an equation. The general form of a sinusoidal function is:

y = A * sin(B * (x - C)) + D

Where:
A is the amplitude (half the distance from the peak to the trough),
B is the frequency (2π divided by the period),
C is the phase shift (horizontal shift),
D is the vertical shift.

Let's analyze the given information and use it to determine the values for each parameter:

1. Amplitude (A):
Given that the water level rises to 7.2 m above average sea level and drops to 7.2 m below average sea level, the total height of the tide is 7.2 m + 7.2 m = 14.4 m. Since the amplitude is half of this total height, the amplitude is 14.4 m / 2 = 7.2 m.

2. Frequency (B):
The time between the peak and the trough is approximately 6 hours. Since one complete cycle of a sinusoidal function consists of a peak, trough, and then back to a peak, the period is 6 hours * 2 = 12 hours. Therefore, the frequency is 2π divided by the period, which is 2π / 12 = π / 6.

3. Phase Shift (C):
At t = 0, the trough of the tide is given as 2 m. The phase shift represents the horizontal displacement of the function. Since at t = 0, the tide is at its trough, the phase shift is 0.

4. Vertical Shift (D):
The vertical shift represents the vertical displacement of the function. The problem does not provide a specific vertical shift value. However, assuming that the average sea level is the center point of the tide, the vertical shift would be at 0.

Now that we have determined the values for A, B, C, and D, we can construct the equation of the sinusoidal function:

y = 7.2 * sin(π/6 * (x - 0)) + 0

Simplifying further:

y = 7.2 * sin(πx/6)

Therefore, the equation of the sinusoidal function that models the given tide situation, assuming that at t = 0 the trough of the tide is 2 m, is y = 7.2 * sin(πx/6).