Chapter 4.8, Problem 69E

a.

To determine

To graph: Graph the data points and model in the same viewing window using graphing utility.

Answer to Problem 69E

  

Explanation of Solution

Given: The numbers of hours $H$ of daylight in Denver, Colorado, on the $15\text{th}$ of each month are given by ordered pairs of the form $\left(t,H\left(t\right)\right)$ , where $t=1$ represents January. A model of the data is

  $H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Data points:

$x$	1	2	3	4	5	6	7	8	9	10	11	12
$y$	9.68	10.73	11.93	13.27	14.37	14.97	14.70	13.73	12.45	11.17	10.00	9.38

Graph:

By using the graphing utility the below graph is the graphical representation of the given data points and the given function.

  

Thus we can use a graphical utility to graph the data points and model in the same viewing window.

b.

To determine

To find: The period of the model.

Answer to Problem 69E

The period of the given model is $12$ and it is the expected result because there are $12$ months in a year.

Explanation of Solution

Given: The numbers of hours $H$ of daylight in Denver, Colorado, on the $15\text{th}$ of each month are given by ordered pairs of the form $\left(t,H\left(t\right)\right)$ , where $t=1$ represents January. A model of the data is$H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Data points:

$x$	1	2	3	4	5	6	7	8	9	10	11	12
$y$	9.68	10.73	11.93	13.27	14.37	14.97	14.70	13.73	12.45	11.17	10.00	9.38

Calculation:

A sine function is given by,

  $y-a\sin \left(bx-c\right)+d$

The period is,

  $\begin{array}{c}\text{Period}=\frac{2\pi }{b}\\ \text{Period}=\frac{2\pi }{\frac{\pi }{6}}\\ \text{Period}=\frac{2\pi \left(6\right)}{\pi }\\ \text{Period}=12\end{array}$

Thus we can find the period of the given model.

c.

To determine

To find: The amplitude of the model.

Answer to Problem 69E

The amplitude of the model is $2.81$ and represents the maximum change in hours of daylight.

Explanation of Solution

Given: The numbers of hours $H$ of daylight in Denver, Colorado, on the $15\text{th}$ of each month are given by ordered pairs of the form $\left(t,H\left(t\right)\right)$ , where $t=1$ represents January. A model of the data is$H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Data points:

$x$	1	2	3	4	5	6	7	8	9	10	11	12
$y$	9.68	10.73	11.93	13.27	14.37	14.97	14.70	13.73	12.45	11.17	10.00	9.38

The amplitude of the sine function is $\left|a\right|$ .

Therefore, the amplitude of the given model is,

  $\left|a\right|=\left|2.81\right|=2.81$

Thus we can find the amplitude of the given model and it represents the maximum change in hours of daylight.