Tide of an island. Find the height of a tide.

Question

Suppose the tide of an island can be modeled by a sinusoidal function. At 4:00 am, there is a high tide of 15 feet. At 10:30 am, there is a low tide of 5 feet. Find the height of the tide at 4:00 pm. (Hint: Find amplitude, period, horizontal shift, and vertical shift first.)

Answer 1

The general form of the sinusoidal function would be

 

f(t) = A sin[B(t - C)] + D

 

 

 

From a high tide at 4 am to a low tide at 10:30 am, the half period is 6.5 hours.  The full period, T = 1/f = 13 hours.

 

B = 2π/T = 2π/13

 

 

 

We want t = 0 at 12 midnight, so when t = 4 for 4 am, the sine function is a maximum for high tide.  sin(π/2) = sin[(2π/13)(4 - C)]

 

Solving for C, C = 4 - 13/4 = 3/4.

 

 

 

A is half of the difference between the maximum and the minimum of f(x): (15 - 5)/2 = 5.

 

 

 

The vertical shift, D, is the number added to A to get the max: 

 

D + 5 = 15

 

D = 10

 

 

 

f(x) = 5 sin[(2π/13)(t - 3/4)] + 10