The water level varies from 10 inches at low tide to 40 inches at high tide. Low tide occurs at 9:15am. and high tide occurs at 3:30pm. What is a cosine function that models variation in inches above and below the water level as a function of time in hours since 9:15 am.?

Answer 1

Let the cosine function be represented as y = a cos(b(x-c)) + d, where a is the amplitude, b is the frequency, c is the phase shift, and d is the vertical shift.

Since the water level varies from 10 to 40 inches, the amplitude is (40-10)/2 = 15 inches.
The period of the function is 6 hours (from low tide at 9:15 am to high tide at 3:30 pm), so the frequency is 2π/6 = π/3.
The phase shift is 9:15 am, which is 15/60 + 9 = 15.25 hours.
The vertical shift is halfway between the high and low tide, which is (40+10)/2 = 25 inches.

Therefore, the cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 am is:
y = 15 cos(π/3(x-15.25)) + 25.