Chapter 4.5, Problem 84E

a.

To determine

Find a trigonometric model for the temperatures in International Falls.

Answer to Problem 84E

  $\boxed{I\left(t\right)=32.4\cos \left(\frac{\pi t}{6}-3.67\right)+46.2}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

Model for the temperatures in Las Vegas is $\text{L}\left(\text{t}\right)=\text{ 8}0.\text{6}0+\text{ 23}.\text{5}0\text{ cos}\left(\pi \text{t 6 }-\text{ 3}.\text{67}\right).$ Find a trigonometric model for the temperatures in International Falls.

Calculation:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas L, and international falls l, in degrees Fahrenheit, for month t,

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

Now to model the data for international Falls,

  $y=a\cos \left(bt-c\right)+d$

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

  $=\frac{1}{2}\left(78.6-13.8\right)$

  $=32.4$

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

  $=2\left(7-1\right)$

  $=12$

  $b=\frac{2\pi }{p}$

  $=\frac{2\pi }{12}$

  $=\frac{\pi }{6}$

Because the maximum temperature occurs in the seventh month,

Therefore,

  $\frac{c}{b}=7$

So, $c=7.67$

The average temperature is

  $=\frac{1}{2}\left(78.6+13.8\right)$

  $=46.2$

So, $d=46.2$

Hence, the model is $\boxed{I\left(t\right)=32.4\cos \left(\frac{\pi t}{6}-3.67\right)+46.2}$

b.

To determine

How well does the model fit the data?

Answer to Problem 84E

  

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

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:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas L, and international falls l, in degrees Fahrenheit, for month t,

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

Let us first plot the points and graph the model Los Vegas,

  

Hence, the graph is plotted.

c.

To determine

How well does the model fit the data?

Answer to Problem 84E

  

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

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:We are given with the data, which shows the maximum daily high temperatures in Las Vegas L, and international falls l, in degrees Fahrenheit, for month t,

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

The model fit the data well.

Let us now plot the points and graphs the model of international falls,

  

Hence, the graph is plotted.

d.

To determine

Use the models to estimate the average maximum temperature in each city.

Answer to Problem 84E

Las Vegas: $\boxed{80.6°}$ , International Falls: $\boxed{46.2°}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

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

Calculation:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas L, and international falls l, in degrees Fahrenheit, for month t,

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

Hence, The annual average temperature is

Las Vegas: $\boxed{80.6°}$

International Falls: $\boxed{46.2°}$

We used the constant term.

e.

To determine

What is the period of each model?

Answer to Problem 84E

  $\boxed{12}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

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

Calculation:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas $\text{L}$ , and international falls l, in degrees Fahrenheit, for month $\text{t},$

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

Hence, The period for both the model is $12$ .

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

f.

To determine

Which city has the greater variability in temperature throughout the year? Which value in each model determines this variability? Explain

Answer to Problem 84E

  $\boxed{\text{International Falls}}$ , $\boxed{\text{Amplitude}}$

Explanation of Solution

Given information:

The table shows the maximum daily high temperatures (in degrees Fahrenheit) in Las Vegas $\text{L}$ and International Falls I for month $\text{t},$ where $\text{t }=\text{ 1}$ corresponds to January. (Source: National Climatic Data Centre)

Which city has the greater variability in temperature throughout the year? Which value in each model determines this variability? Explain

Calculation:

We are given with the data, which shows the maximum daily high temperatures in Las Vegas $\text{L}$ , and international falls l, in degrees Fahrenheit, for month $\text{t},$

  $\begin{array}{ccc}Month,t& Las-vagas,L& International-Falls,l\\ 1& 57.1& 13.8\\ 2& 63.0& 22.4\\ 3& 69.5& 22.4\\ 4& 78.1& 51.5\\ 5& 87.8& 66.6\\ 6& 98.9& 74.2\\ 7& 104.1& 78.6\\ 8& 101.8& 76.3\\ 9& 93.8& 64.7\\ 10& 80.8& 51.7\\ 11& 66.0& 32.5\\ 12& 57.3& 18.1\end{array}$

We are given with a model for the temperature in Las Vegas,

  $L\left(t\right)=80.60+23.50\cos \left(\frac{\pi t}{6}-3.67\right)$

Hence , International Falls has the greater variability,

Amplitude determines this factor.

The greater the amplitude, greater is the variability.