Answer
a. $10\ ft$ at 6 A.M., $14\ ft$ at 9 A.M., $10\ ft$ at 6 P.M., $10\ ft$ at midnight, $6\ ft$ at 3 A.M.
b. low tide 3 P.M. and 3 A.M.; high tide 9 A.M. and 9 P.M.
c. $12\ hours$, see explanations.
Work Step by Step
a. With the given function $H(t)=10+4sin(\frac{\pi}{6}t)$, we have:
$H(0)=10+4sin(0)=10\ ft$ at 6 A.M., $H(3)=10+4sin(\frac{\pi}{2})=14\ ft$ at 9 A.M., $H(12)=10+4sin(2\pi)=10\ ft$ at 6 P.M., $H(18)=10+4sin(3\pi)=10\ ft$ at midnight, $H(21)=10+4sin(\frac{7\pi}{2})=6\ ft$ at 3 A.M.
b. Based on the above results, we can identify that low tide happens at 3 P.M. and 3 A.M., while high tide happens at 9 A.M. and 9 P.M.
c. The period of this function can be found as $p=\frac{2\pi}{\pi/6}=12\ hours$, which means that the low and high tides will repeat every 12 hours.
