At the Bay of Fundy, the difference between high and low tide is 8m and the time between high tides is 12 hours. At midnight, the water is at high tide. Write a sine function that models the water level over time t.
To write a sine function that models the water level over time t, we need to consider two factors: the amplitude and the period.
Given that the difference between high and low tide is 8m, the amplitude of the sine function is half of this value. So the amplitude, A, is 8m/2 = 4m.
Given that the time between high tides is 12 hours, the period of the sine function is also 12 hours. The period, P, is equal to 2π divided by the frequency, f. Since the frequency is the inverse of the period, the frequency, f, is equal to 1 divided by 12 hours.
Therefore, the period, P, is 2π/(1/12) = 24π hours.
Now, we can write the sine function:
h(t) = A * sin(2πft/24π) = 4 * sin((2π/12) * t)
To write a sine function that models the water level over time at the Bay of Fundy, we need to consider the amplitude, period, and phase shift.
1. Amplitude: The amplitude is half the difference between the high and low tide. In this case, the amplitude is 8m/2 = 4m.
2. Period: The period is the time it takes for one complete cycle. In this case, the time between high tides is 12 hours, so the period is 12 hours.
3. Phase shift: The phase shift determines the horizontal shift of the sine function. Since at midnight the water is at high tide, we consider this as the starting point (phase shift = 0).
Based on these values, the sine function that models the water level over time t can be written as:
y = A sin(Bx + C) + D
where:
A = amplitude = 4m,
B = 2π/period = 2π/12 = π/6,
C = phase shift = 0,
D = vertical shift = average water level = (high tide + low tide) / 2 = (0 + 8) / 2 = 4m.
Therefore, the sine function that models the water level over time t is:
y = 4 sin(π/6x) + 4.