Given:
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.
Required:
We need to determine the equation of a sinusoidal function that would model this situation.
Explanation:
Let t be the time in seconds.
let h be the height of the waves in meters.
Consider the sinusoidal function.
[tex]h=Asin(Bt)+C[/tex]where A is the amplitude, B is the period of the function, and C is the y-intercept.
Consider the drawing of the given situation.
We know that the amplitude of a function is the amount by which the graph of the function travels above and below its midline.
The midline is the x-axis.
The wave travels 2 m above the midline so the amplitude is 2.
[tex]A=2[/tex]Given that at equation t=0, the propeller is at its resting position.
We know that the height at the rest position is 0.
We get (h,t)=(0.0).
The point (0,0) is the y-intercept of the given function since the function meets the y-axis at (0,0).
[tex]C=0[/tex]We know that the time interval between two waves is known as a Period.
The time between the peak and the trough is the period.
The period of the given function is 3.
[tex]The\text{ period of the sine funciton is }2\pi.[/tex][tex]\frac{2\pi}{B}=3[/tex][tex]\frac{2\pi}{3}=B[/tex][tex]Substitute\text{ }A\text{ =2, B=}\frac{2\pi}{3},\text{ and C=0 in the sinusoidal function.}[/tex][tex]h=2sin(\frac{2\pi}{3}t)+0[/tex]Final answer:
[tex]h=2sin(\frac{2\pi}{3}t)[/tex]
