Question

The water level varies from 12 inches at low tide to 52 inches at high tide. Low tide occurs at 9:15 a.m. and high tide occurs at 3:30 p.m. What is a cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m.? show all work in short terms

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find a cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m., we can use the formula for a cosine function:

h(t) = A * cos(B(t - C)) + D

Where:
- A is the amplitude, which is half the difference between the maximum and minimum values, so A = (52 - 12) / 2 = 20
- B is the period, which is 2π divided by the length of one complete cycle, so B = 2π / 6.25 = 0.8π
- C is the horizontal shift, which is the time at which the function reaches its maximum value, so C = 3.25
- D is the vertical shift, which is the average of the maximum and minimum values, so D = (52 + 12) / 2 = 32

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 a.m. is:

h(t) = 20 * cos(0.8π(t - 3.25)) + 32