Example 5 from Precalculus, 7.6 Modeling with Trigonometric Functions

Describing Periodic Motion

The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let $y$ equal the distance from the tip of the hour hand to the ceiling $x$ hours after noon. Find the equation that models the motion of the clock and sketch the graph.

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
Begin by making a table of values as shown in Table 4.

$x$ $y$ Points to plot

Noon 30 in $(0, 30)$

3 PM 54 in $(3, 54)$

6 PM 78 in $(6, 78)$

9 PM 54 in $(9, 54)$

Midnight 30 in $(12, 30)$

Table 4

To model an equation, we first need to find the amplitude.

$\begin{array}{l} \begin{array}{l} | A | = | \frac{78 - 30}{2} | \\ = 24 \end{array} \end{array}$
The clock’s cycle repeats every 12 hours. Thus,

$\begin{array}{l} \begin{array}{l} B = \frac{2 π}{12} \\ = \frac{π}{6} \end{array} \end{array}$

The vertical shift is

$\begin{array}{l} \begin{array}{l} D = \frac{78 + 30}{2} \\ = 54 \end{array} \end{array}$

There is no horizontal shift, so $C = 0.$ Since the function begins with the minimum value of $y$ when $x = 0$ (as opposed to the maximum value), we will use the cosine function with the negative value for $A .$ In the form $y = A \cos (B x \pm C) + D,$ the equation is

$y = −24 \cos (\frac{π}{6} x) + 54$
See Figure 9.

Figure 9

$x$	$y$	Points to plot
Noon	30 in	$(0, 30)$
3 PM	54 in	$(3, 54)$
6 PM	78 in	$(6, 78)$
9 PM	54 in	$(9, 54)$
Midnight	30 in	$(12, 30)$

How did it go?

Solution

Solution