Chapter 4.8, Problem 66E

(a).

To determine

To graph:

The data points and the model in the same viewing window.

Explanation of Solution

Given:

The numbers of hours $H$ of daylight in Denver, Colorado, on the $15th$ 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 for the data is $H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Graph:

  

(b).

To determine

To find:

The period of the model.

Answer to Problem 66E

The period is $12$ . Yes, the period and the months in a year is the same.

Explanation of Solution

Given:

The numbers of hours $H$ of daylight in Denver, Colorado, on the $15th$ 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 for the data is $H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Calculation:

The sine function is:

  $y=a\sin \left(bx-c\right)+d$ and the period of this function is given by $\frac{2\pi }{b}$ .

According to the question $y=2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

So, substitute $b=\frac{\pi }{6}$ in a period formula:

  $\begin{array}{c}Period=\frac{2\pi }{\frac{\pi }{6}}\\ =\frac{12\pi }{\pi }\\ =12\end{array}$

Also, from the year calendar, there are $12$ months in a year.

Yes, the period and the months in a year is the same.

Conclusion:

Hence, the period is $12$ . Yes, the period and the months in a year is the same.

(c).

To determine

To find:

The amplitude of the model.

Answer to Problem 66E

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

Explanation of Solution

Given:

The numbers of hours $H$ of daylight in Denver, Colorado, on the $15th$ 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 for the data is $H\left(t\right)=12.18+2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

Calculation:

The sine function is:

  $y=a\sin \left(bx-c\right)+d$ and the amplitude is $\left|a\right|$ .

According to the question $y=2.81\sin \left(\frac{\pi t}{6}-\frac{\pi }{2}\right)$ .

So, substitute $a=2.81$ in the amplitude formula:

  $\begin{array}{c}Amplitude=\left|2.81\right|\\ =2.81\end{array}$

This amplitude represents the maximum change in hours of daylight.

Conclusion:

Hence, the amplitude is $2.81$ and this amplitude represents the maximum change in hours of daylight.