Question

A buoy starts at a height of 0 in relation to sea level and then goes up. Its maximum displacement in either direction is 6 feet, and the time it takes to go from its highest point to its lowest point is 4 seconds. Which equations can be used to model h, the height in feet of the buoy in relation to sea level as a function of time, t, in seconds?

Answer 1

Answer:

$h=6*sin(\frac{2\pi }{8}t)$

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 $t=0$, $h=0$, this tells us that trigonometric function modeling the buoy must be the sine function since for

$h=A*sin(wt)$, at   $t=0$, $h=0$.

The maximum displacement of the buoy in either direction is 6 feet, this means we have

$h=6*sin(wt)$

now we need to figure out $w$.

The time it takes the buoy to get from its lowest point to its highest point is 4 seconds, that means the period $T$ of oscillation is $2*4=8$ 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 $h=6*sin(wt)$ the we have:

$wT=2\pi$

$w(8)=2\pi \\\\\boxed{w=\frac{2\pi }{8} }$

Now the final equation looks like

$h=6*sin(\frac{2\pi }{8}t)$

Answer 2

Answer:

[C] h = 6sin $(\frac{\pi }{4} t)$