The depth of the water at the end of a pier changes periodically along with the movement of tides. on a particular day, low tides occur at 12: 00 am and 12: 30 pm, with a depth of 2.5 m, while high tides occur at 6: 15 am and 6: 45 pm, with a depth of 5.5 m. let t = 0 be 12: 00 am. write a cosine model, d = acos(bt) + k, for the depth as a function of time. this amplitude is meters. a =

Draw a diagram to illustrate the problem as shown in the figure below.

The minimum depth of 2.5 m occurs at 12:00 am and at 12:30 pm.
Therefore the period i0s T= 12.5 hours.
The maximum depth of 5.5 m occurs at 6:15 am and at 6:45 pm. Therefore the period of T = 12.5 hours is confirmed.

The double amplitude is 5.5 - 2.5 = 3 m, therefore the amplitude is a = 1.5 m.
The mean depth is k = (2.5 + 5.5)/2 = 4.0 m

The model for tide depth is


That is,
d = -1.5 cos(0.5027t) + 4
where 
d = depth, m
t = time, hours

A plot of the function confirms that the model is correct.