Example 6 from Precalculus, 7.6 Modeling with Trigonometric Functions

Determining a Model for Tides

The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as

$\begin{array}{l} \begin{array}{l} | A | = | \frac{(15 - 7)}{2} | \\ = 4 \end{array} \end{array}$

The cycle repeats every 12 hours; therefore, $B$ is

$\frac{2 π}{12} = \frac{π}{6}$

There is a vertical translation of $\frac{(15 + 7)}{2} = 11.5.$ Since the value of the function is at a maximum at $t = 0,$ we will use the cosine function, with the positive value for $A .$
$y = 4 \cos (\frac{π}{6}) t + 11$
See Figure 10.

Figure 10

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The daily temperature in the month of March in a certain city varies from a low of $24 °F$ to a high of $40 °F .$ Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point $32 °F .$ Let $t = 0$ correspond to noon.

How did it go?

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The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

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Solution

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