Chapter 4.5, Problem 89E

a.

To determine

Find a trignometric model for International Falls.

Answer to Problem 89E

  $\boxed{I(t)=32.4\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)+46.2}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Find a trignometric model for International Falls.

Calculation:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ , in degrees Fahrenheit, for a month $t$ .

  

We are given with the model of temperature in Las Vegas.

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Now to model the data for International Falls,

  $y=a\cos (bt-c)+d$

Amplitude $a=1/2$ [max temp.-min temp.]

  $\begin{array}{l}=\begin{array}{c}1\\ 2\end{array}(78.6-13.8)\\ =32.4\end{array}$

Period $p=2$ [month of max temp.-month of min temp.]

  $\begin{array}{l}=2(7-1)\\ =12\end{array}$

Also,

  $\begin{array}{l}b=\begin{array}{c}2\pi \\ p\end{array}\\ =\begin{array}{c}2\pi \\ 12\end{array}=\begin{array}{c}\pi \\ 6\end{array}\end{array}$

Because maximum temperature occurs in seventh month,

So,

  $\begin{array}{l}\begin{array}{c}c\\ b\end{array}=7\\ c=7.67\end{array}$

The average temperature is,

  $\begin{array}{l}d=\begin{array}{c}1\\ 2\end{array}(78.6+13.8)\\ d=46.2\end{array}$

Hence, the model is $\boxed{I(t)=32.4\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)+46.2}$

b.

To determine

Graph the data points.

How well does the model fit the data?

Answer to Problem 89E

  

The model fits the data well.

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Use a graphing utility to graph the data points and the model for the temperatures in Las Vegas. How well does the model fit the data?

Calculation:

Let us first plot the plots and graph the model of Las Vegas.

  

Hence, the model fits the data well.

c.

To determine

Graph the data points and the model for the temperatures in International Falls.

Answer to Problem 89E

  

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Use the graphing utility to graph the data points and the model for the temperatures in International Falls. How well does the model fit the data?

Calculation:

Let us first plot the plots and graph the model of International Falls.

  

Hence, the result.

d.

To determine

Average maximum temperature.

Answer to Problem 89E

  $\boxed{{80.6}^{\circ }}$

  $\boxed{{46.2}^{\circ }}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Use the models to estimate the average maximum temperature in each city. Which term of the models did you use? Explain.

Calculation:

The annual average temperature is,

Las Vegas: $\boxed{{80.6}^{\circ }}$

International Falls: $\boxed{{46.2}^{\circ }}$

Hence, average temperature of Las Vegas is $\boxed{{80.6}^{\circ }}$ and International Falls is $\boxed{{46.2}^{\circ }}$ .

e.

To determine

Period of each model.

Answer to Problem 89E

  $\boxed{12}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

What is the period of each model? Are the periods what you expected? Explain.

Calculation:

The period for both the model is $12$ .

Yes, this is what we expected, as the number of months in a year is $12$ .

Hence, the period for both the model is $\boxed{12}$ .

f.

To determine

Which city has the greater variability in temperature throughout the year?

Factor of the models determines this variability.

Answer to Problem 89E

International Falls.

Amplitude.

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month with corresponding to January.

A model for the temperatures in Las Vegas is

  $L(t)=80.60+23.50\cos (\begin{array}{c}\pi t\\ 6\end{array}-3.67)$

Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain

Calculation:

International Falls has greater variability,

Amplitude determines this factor.

Hence, greater the amplitude, greater is the variabilty.