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.?

52 inches - 12 inches = 40 inches

amplitude: a = 40 inches / 2 = 20

f (x) = 20cos (bx) + c

the value of c is 32 ... since the centre of the has been moved up 32 units

the minimum amplitude = 32 - 20 = 12

the maximum amplitude = 32 + 20 = 52

f (x) = 20cos (bx) + 32

if the curve takes 6 1/4 hours from low to high tides (9:15 am to 3:30 pm) then it will take 12 1/2 hours to complete a full cycle.

adjust the period by converting 12 1/2 hours to an angle measure.

360 °/12 = 30°

30° / 12 = 15°

12 1/2 = 360° + 15° = 375°

f (x) = 20 cos (375°) + 32

f (x) = 20 * 0.97 + 32

f (x) = 19.4 + 32

f (x) = 51.4