The propeller of a boat at dock in the ocean will rise and fall with the waves. On a particularly wavy night, the propeller leaves its resting position and reaches a height of 2m on the peaks of the waves and -2m in the troughs. The time between the peak and the trough is approximately 3 seconds. Determine the equation of a sinusoidal function that would model this situation assuming that at equation , the propeller is at its resting position and headed towards the peak of the next wave.
h=2 m. = Ht. or amplitude of the wave.
P = 2 * 3s = 6 s. = The period or time
for i cycle.
F = 1/P = 1/6 c/s = 1/6 Hz = Frequency of the wave.
h = hmax*sinWt.
h = 2*sinWt.
W = 2pi*F = 2piRad/c * (1/6)c/s = 2pi/6 = pi/3 Rad/s.
Eq: h = 2*sin((pi/3)t)
To determine the equation of a sinusoidal function that models the propeller's motion, we need to consider the properties of a sinusoidal wave, specifically amplitude, period, and vertical shift.
1. Amplitude: The amplitude is the height from the resting position to the peak (or trough) of the wave. In this case, the amplitude is 2 meters, since the propeller reaches a height of 2m on the peaks and -2m on the troughs.
2. Period: The period of a sinusoidal wave is the time it takes to complete one full cycle. In this case, the time between the peak and the trough is 3 seconds. Since a full cycle includes both the peak and the trough, the period is twice this value, which is 6 seconds.
3. Vertical Shift: The vertical shift represents any upward or downward displacement from the resting position. In this case, the resting position is at 0m, so there is no vertical shift.
Now we can write the equation of the sinusoidal function using the general form:
f(x) = A*sin(Bx - C) + D
where:
A = amplitude
B = 2π / period
C = phase shift
D = vertical shift
Plugging in the values we determined:
A = 2 (amplitude)
B = 2π / 6 = π / 3 (period)
C = 0 (no phase shift)
D = 0 (no vertical shift)
The equation becomes:
f(x) = 2*sin((π/3)x)
Therefore, the equation of the sinusoidal function that models the propeller's motion is f(x) = 2*sin((π/3)x).
To determine the equation of a sinusoidal function that models this situation, we need to identify the important parameters of the wave: amplitude, period, phase shift, and vertical shift.
The amplitude of the wave is the distance between the resting position and the peak (or trough) of the wave. In this case, the amplitude is 2m.
The period of the wave is the time it takes for a complete cycle (from one peak to the next or from one trough to the next). In this case, the period is approximately 3 seconds.
The vertical shift represents the resting position of the wave. In this case, the resting position is at 0m.
Lastly, the phase shift determines the horizontal position of the wave. Since the propeller is at its resting position and headed towards the peak of the next wave, there is no phase shift.
With these parameters, the equation of the sinusoidal function can be written as:
y = A*sin(Bx) + C
Where:
A is the amplitude
B = 2π / period
C is the vertical shift
Substituting the known values, we have:
y = 2*sin((2π/3)x) + 0
Therefore, the equation of the sinusoidal function that models this situation is:
y = 2*sin((2π/3)x)