Answer:
[tex]h=6*sin(\frac{2\pi }{8}t)[/tex]
Step-by-step explanation:
Since the buoy is going up and down, its motion is oscillatory, and therefore, it can be described by a trigonometric function.
The buoy starts at height of 0 in relation to the sea level, which means at time [tex]t=0[/tex], [tex]h=0[/tex], this tells us that trigonometric function modeling the buoy must be the sine function since for
[tex]h=A*sin(wt)[/tex], at [tex]t=0[/tex], [tex]h=0[/tex].
The maximum displacement of the buoy in either direction is 6 feet, this means we have
[tex]h=6*sin(wt)[/tex]
now we need to figure out [tex]w[/tex].
The time it takes the buoy to get from its lowest point to its highest point is 4 seconds, that means the period [tex]T[/tex] of oscillation is [tex]2*4=8[/tex] seconds. The buoy returns to to the same place after one period, this means if the buoy started from 0, it will return to 0 after 8 seconds; therefore, from [tex]h=6*sin(wt)[/tex] the we have:
[tex]wT=2\pi[/tex]
[tex]w(8)=2\pi \\\\\boxed{w=\frac{2\pi }{8} }[/tex]
Now the final equation looks like
[tex]h=6*sin(\frac{2\pi }{8}t)[/tex]
