A buoy, bobbling up and down in the water, moves from its highest to lowest and back to highest point in 14 seconds. The distance between high and low is 5 ft. find the sinusoid function for the this situation.
Since the buoy moves from highest to the next highest point in 14 seconds, the period of oscillation is 14 seconds.
The distance from maximum to minimum height is 5ft. so the amplitude is 2.5ft. (The midline is located midway from the highest and lowest point; the amplitude is the maximal distance from the midline.)
We are not given a starting time, nor the average depth of the water so we are not concerned with horizontal displacement (phase shift) nor vertical translation. Also, we can use either sine or cosine.
Using sine the general form is y=Asin(B(x-h))+k with A the amplitude, B affected by the period, h the horizontal translation (phase shift) and k the vertical translation.We have the amplitude A=2.5. Since the period p is 14, `B=(2pi)/p=(2pi)/14=pi/7 ` .
The model is `y=2.5sin((pi t)/7) ` where t is time in seconds.