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.?

Answer 1

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