Chapter 13.9, Problem 52STP

To determine

To write an equation that represents the height of the tide h.

Answer to Problem 52STP

  $h=27.3\sin \frac{\pi }{6}t$

Explanation of Solution

Given: The world record for the highest tide is held by the Minas Basin with a tidal range of 54.6 feet. A tide is said to be in equilibrium when it is at its normal level halfway between its highest and lowest points. The tide completes one cycle in 12 hours. An equation to represent the height of the tide h is to be written. The tide completes one cycle in 12 hours and it is at equilibrium at t = 0 and the high tide is beginning.

Concept used: The general form of a wave/tide equation which has equilibrium at t = 0 is $y=A•\sin \frac{2\pi }{b}t$ , where A is the amplitude and t is the time and $\frac{2\pi }{b}$ is equal to the period of one cycle.

The amplitude of the tide = half of the range = $\frac{54.6}{2}=27.3$ .

Period is 12 hours, which is equal to $\frac{2\pi }{b}$ . Now,

  $\begin{array}{l}\frac{2\pi }{b}=12\\ \Rightarrow b=\frac{2\pi }{12}\\ \Rightarrow b=\frac{\pi }{6}\end{array}$

The required function will be sin function because it has the quality of equilibrium at t = 0.

Hence the required equation of the tide;

  $h=27.3\sin \frac{\pi }{6}t$