Given the function:
[tex]f(x)=5cos\frac{\pi}{6}t+4[/tex]The t is the number of hours after midnight.
Let's estimate how many hours elapse between the first two times the buoy is exactly 6 m above sea level.
We have:
[tex]f(x)=6[/tex]Substitute 6 for f(x) and solve for t.
We have:
[tex]\begin{gathered} 6=5cos\frac{\pi}{6}t+4 \\ \\ 6-4=5cos\frac{\pi}{6}t+4-4 \\ \\ 2=5cos\frac{\pi}{6}t \end{gathered}[/tex]Divide both sides by 5:
[tex]\begin{gathered} \frac{2}{5}=\frac{5cos\frac{\pi}{6}t}{5} \\ \\ \frac{2}{5}=cos\frac{\pi}{6}t \end{gathered}[/tex]Take the cos inverse of both sides:
[tex]\begin{gathered} cos^{-1}(\frac{2}{5})=\frac{\pi}{6}t \\ \\ 66.422=\frac{\pi}{6}t_1 \\ \\ \frac{\pi}{6}t_1=66.42 \end{gathered}[/tex]Also, the second angle is:
[tex]\begin{gathered} \frac{\pi}{6}t=360-66.42 \\ \\ \frac{\pi}{6}t_2=293.58 \end{gathered}[/tex]Now, let's solve both equations for t:
[tex]\begin{gathered} t_1=66.42*\frac{6}{\pi}=\frac{66.42*6}{180}=2.214 \\ \\ t_2=\frac{293.58*6}{180}=9.786 \end{gathered}[/tex]First time = 2.214 hours
Second time = 9.786 hours
Now, to solve for the time, t, we have:
t = t2 - t1 = 9.786 - 2.214 = 7.575 ≈ 7.6 hours.
Therefore, 7.6 hours elapsed between the first two times the buoy is 6m above sea level.
ANSWER:
D. 7.6 hours.
