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

F ( x ) = a cos ( b ( x - c ) ) + d
Apmlitude: a = ( 12 - 52 ) / 2 = -40 / 2 = - 20
If the curve takes 6.25 hours from the low to the high tide them it will take 12.50 hours to complete a full cycle.
The period:   b = 2π/12.5 ;
c = 9.25; d = 12 + 20 = 32;
Answer:
f ( x ) = - 20 cos ( 2π/12.5 ·( x - 9.25 ) ) + 32 =
= - 20 cos ( 0.5024 ( x - 9.25 ) ) +32