Question

Due to the high and low tides, the depth d of water in a certain costal area may be expressed by a sinusoidal function. The highest tide occurs at 8 am and the lowest tide occurs 6 hours later. The maximum level of water is 2.8 meters and the lowest level of water is 0.4 meters. 
a) Use sinusoidal functions to find the depth d(t) of the water, in meters, as a function of time t in hours. (Assume that 8 am corresponds to t = 0). 
b) Find the depth of water at noon. 
c) Use the graph of d(t) and analytical calculations to calculate the interval of time during which the depth d is below 1.5 m from 12 pm to 6 pm. ​

Answer 1

Solution

a) Let d(t) be written as

d(t) = a cos[b(t - d)] + c

The minimum dmin and the maximum dmax of d are

dmin = 0.4

dmax = 2.8

c = (dmax + dmin) / 2 = (2.8 + 0.4) / 2 = 1.6

|a| = (dmax - dmin) / 2 = (2.8 - 0.4) / 2 = 1.2

Since d(t) has a minimum at t = 0 (8 am), we can select a = - 1.2 and d = 0

d(t) = -1.2 cos[b(t)] + 1.6 , where it is easy to check that d = -1.2 + 1.6 = 0.4 at t = 0.

We now use the period to find b (b > 0) as follows

period = 12 = 2π / b

hence b = π / 6

d(t) is now written as

d(t) = -1.2 cos[ (π / 6)(t) ] + 1.6 , where it is easy to check that the maximum occurs at t = 6.

b) At noon t = 4, hence

d(4) = -1.2 cos[ (π / 6)(4) ] + 1.6 = 2.2 m

c) The graph of d(t) is shown below with vertical lines corresponding to 12 pm to 6 pm and a horizontal line corresponding to d = 1.5. The depth of the water is less than 1.5 from t0 to 6 pm (t = 10). We need to find t0 which is a point of intersection of d(t) and y = 1.5 by solving the equation

-1.2 cos[ (π / 6)(t) ] + 1.6 = 1.5

Graph of y = d(t) and y = 1.5

t = 6 arccos ( (1.5 - 1.6)/(-1.2) ) / π ≈ 2.84

The solution found corresponds to the left point of intersection of d(t) and y = 1.5. The right point t0 may be found using the symmetry of the graph with respect to the maximum point. Hence

t0 = 6 + (6 - 2.84) = 9.16 hours

9.16 hours corresponds to 5 pm and 0.16×60 minutes ≈ 5:10 pm

From 12 pm to 6 pm, the depth of water is below 1.5 from 5:10 pm to 6pm