Chapter 12.8, Problem 4GP

To determine

To find: Write a cosine function that represents the height of water.

Answer to Problem 4GP

Height of the water is represented by the function

  $h=4\cos \frac{\pi }{6}t+10$

Explanation of Solution

Given information:

Height of water in a wave pool oscillates between a maximum of $14$ feet and a minimum of $6$ feet.

Wave generator pumps $5$ waves per minute.

First we need to write the equation for the midline, and determine the vertical shift.

  $y=\frac{14+6}{2}=10$

So, the midline is $y=10$ , the vertical shift is $k=10$ .

Now we need to find amplitude.

Amplitude is the difference between the midline value and the maximum value.

  $\left|a\right|=\left|14-10\right|=4$

So, amplitude is $\left|a\right|=4$

Now we need to find period

Since there are $5$ waves per minute, there is $1$ wave for every $\frac{60}{5}$ seconds or $12$ seconds.

So, the period is $12$ seconds.

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

Write an equation for the function

  $h=a\cos b\left(t-h\right)+k$

On substituting values

We get

  $\begin{array}{l}h=a\cos b\left(t-h\right)+k\\ \Rightarrow h=4\cos \frac{\pi }{6}\left(t-0\right)+10\\ \Rightarrow h=4\cos \frac{\pi }{6}t+10\end{array}$

Therefore,

Height of the water is represented by the function

  $h=4\cos \frac{\pi }{6}t+10$

Graph of the function is