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 12.8 h. How long does it take for the water to fall a distance 0.250d from its highest level

2.13 h

Explanation:

Simple harmonic motion is:

x = A sin (2π/T t + φ) + B

where A is the amplitude, T is the period, φ is the phase shift, and B is the midline.

This can also be written in terms of cosine:

x = A cos (2π/T t + φ) + B

Here, A = d/2, T = 12.8, φ = 0, and B = d/2. I'll use cosine so that the highest level is at t=0.

x = d/2 cos (2π/12.8 t) + d/2

When x = d - 0.250 d = 0.750 d:

0.750 d = d/2 cos (2π/12.8 t) + d/2

0.250 d = d/2 cos (2π/12.8 t)

0.500 = cos (2π/12.8 t)

π/3 = 2π/12.8 t

12.8/6 = t

t = 2.13

It takes 2.13 hours to fall 0.250 d from the highest level.