Question

At a certain harbor, the tides cause the ocean surface to rise and fall a distance d (from highest level to lowest level) in simple harmonic motion, with a period of 12.4 h. how long does it take for the water to fall a distance 0.250d from its highest level?

Answer 1

T = 12.4 h, x(t) = .250 d (displacement)
The total amplitude is half of the distance d from the highest to lowest level. 
Using the equation x(t) = x(m) cos(wt + φ), we get: 

.250d = .5d cos(wt + φ) 
To find w, use the equation w = 2π / T 

.250d = .5d cos((2π / 12.4)t + φ) 

.250d = .5d cos((2π / 12.4)t + 0) 

Solve the equation for t. 
.250d / .5d = cos ((2π / 12.4) t) 
arccos(.5) = 2π / 12.4 * t 
arccos(.5) * 12.4 / 2 / π = t 
t = 2.066666 hours