On Dolphin Beach, the high tide is 2.2 meters and only occurs at 12 a.m. and 12 p.m. The low tide is 1 meter and only occurs at 6 a.m. and 6 p.m.
Which function models the height of the tide t hours after 12 a.m.?
a. h(t)=0.6cos(πt/6)+1.6*
b. h(t)=0.6sin(πt/6)+1.6
c. h(t)=1.2cos(πt/3)+1
d. h(t)=1.6sin(πt/3)+2.2
To determine which function models the height of the tide t hours after 12 a.m., we need to analyze the given information.
Firstly, we are told that the high tide occurs at 12 a.m. and 12 p.m. This means that the maximum height of the tide is 2.2 meters.
Secondly, the low tide occurs at 6 a.m. and 6 p.m., with a height of 1 meter.
From this information, we can infer that the tide goes through one complete cycle every 12 hours, as it goes from high tide (2.2 meters) to low tide (1 meter) and back to high tide again.
Since the sine and cosine functions have a period of 2π, and we need a period of 12 hours for the tide cycle, we need to adjust the argument of the trigonometric function. The correct function that models the tide height is one that has a period of 12 hours, amplitude of (2.2 - 1)/2 = 0.6, and an average value of (2.2 + 1)/2 = 1.6.
Looking at the answer choices, the only option that satisfies these requirements is:
b. h(t) = 0.6sin(πt/6) + 1.6
So, the function that models the height of the tide t hours after 12 a.m. on Dolphin Beach is h(t) = 0.6sin(πt/6) + 1.6.