Question

The tides around Cherokee Bay range between a low of 1 foot to a high of 5 feet. The tide is at its lowest point when time, t, is 0 and completes a full cycle over a 24 hour period. What is the amplitude, period, and midline of a function that would model this periodic phenomenon?

Answer 1

Given:
Low tide height = 1 ft
High tide height = 5 ft
Tide period, T = 24  houts

Let the height of the tide be modeled by the expression
h(t) = K + A cos(bt)
Because the period is 24, therefore
b = (2π)/24 = π/12

That is,
h(t) = K + Acos[(πt)/12]

When r=0, h = 1, therefore
K + A cos(0) = 1, ot
 K + A = 1             (1)
When t = 12 (half cycle), h = 5, therefore
K + A cos(π) = 5, or
K - A = 5             (2)

Add (1) and (2):
2K = 6
K = 3
From(1), obtain
A = 1 - 3 = - 2

Answer:
The required function is h(t) = 3  - 2 cos[(πt)/12]
The amplitude is 2 feet
The period is 24 hours
The midline of the function is h = 3 feet
A graph of the function is shown below.