To determine which of the given equations models the height, [tex]\( h \)[/tex], of the buoy in relation to sea level as a function of time, [tex]\( t \)[/tex], we need to consider the characteristics of the buoy's motion.
1. Amplitude:
The maximum displacement of the buoy from the sea level is 6 feet. Therefore, the amplitude, [tex]\( A \)[/tex], of the sinusoidal function should be 6. This means that in our equation, the coefficient in front of the sine function should be 6.
2. Period:
The buoy takes 4 seconds to go from its highest point to its lowest point. This is half of the period of the sine function (since a full cycle goes from highest point to lowest point and back to the highest point). Therefore, the full period [tex]\( T \)[/tex] of the function is 8 seconds.
3. Form of the equation:
The general form of the sinusoidal function is:
[tex]\[
h(t) = A \sin(B \cdot t)
\][/tex]
where [tex]\( A \)[/tex] is the amplitude and [tex]\( B \)[/tex] is related to the period by:
[tex]\[
B = \frac{2\pi}{T}
\][/tex]
4. Calculate [tex]\( B \)[/tex]:
Since the period [tex]\( T \)[/tex] is 8 seconds, we have:
[tex]\[
B = \frac{2\pi}{8} = \frac{\pi}{4}
\][/tex]
So our equation should be:
[tex]\[
h(t) = 6 \sin\left(\frac{\pi}{4} t\right)
\][/tex]
Now, let's compare this with the given options:
1. [tex]\( h = 4 \sin \left(\frac{\pi}{6} t\right) \)[/tex]
2. [tex]\( h = 4 \sin \left(\frac{\pi}{3} t\right) \)[/tex]
3. [tex]\( h = 6 \sin \left(\frac{\pi}{4} t\right) \)[/tex]
4. [tex]\( h = 6 \sin \left(\frac{\pi}{2} t\right) \)[/tex]
The correct equation that matches our parameters (amplitude 6 and [tex]\(B = \frac{\pi}{4}\)[/tex]) is:
[tex]\[
\boxed{3}
\][/tex]
Hence, the equation [tex]\( h(t) = 6 \sin\left(\frac{\pi}{4} t\right) \)[/tex] correctly models the height of the buoy in terms of time.
