Chapter 6, Problem 95RE

a.

To determine

Draw the scatter diagram of the data for one period.

Answer to Problem 95RE

  

Explanation of Solution

Given information:

The following data represent average monthly temperatures for phoenix, Arizona.

  $\begin{array}{ll}Month,m\hfill & MonthlyAverageTemperarure,t\hfill \\ January,1\hfill & 51\hfill \\ February,2\hfill & 55\hfill \\ March,3\hfill & 63\hfill \\ April,4\hfill & 67\hfill \\ May,5\hfill & 77\hfill \\ June,6\hfill & 86\hfill \\ July,7\hfill & 90\hfill \\ August,8\hfill & 90\hfill \\ September,9\hfill & 84\hfill \\ October,10\hfill & 71\hfill \\ November,11\hfill & 59\hfill \\ December,12\hfill & 52\hfill \end{array}$

Calculation:

Here we have:

  $\begin{array}{ll}{x}_1\hfill & y_1\hfill \\ 1\hfill & 51\hfill \\ 2\hfill & 55\hfill \\ 3\hfill & 63\hfill \\ 4\hfill & 67\hfill \\ 5\hfill & 77\hfill \\ 6\hfill & 86\hfill \\ 7\hfill & 90\hfill \\ 8\hfill & 90\hfill \\ 9\hfill & 84\hfill \\ 10\hfill & 71\hfill \\ 11\hfill & 59\hfill \\ 12\hfill & 52\hfill \end{array}$

Hence, the scatter diagram is as follows:

  

Hence, the result is shown in the above diagram.

b.

To determine

Find a sinusoidal function $y=A\sin \left(\omega x-\varphi \right)+B$ that fits the data.

Answer to Problem 95RE

  $y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5$

Explanation of Solution

Given information:

The following data represent average monthly temperatures for phoenix, Arizona.

  $\begin{array}{ll}Month,m\hfill & MonthlyAverageTemperarure,t\hfill \\ January,1\hfill & 51\hfill \\ February,2\hfill & 55\hfill \\ March,3\hfill & 63\hfill \\ April,4\hfill & 67\hfill \\ May,5\hfill & 77\hfill \\ June,6\hfill & 86\hfill \\ July,7\hfill & 90\hfill \\ August,8\hfill & 90\hfill \\ September,9\hfill & 84\hfill \\ October,10\hfill & 71\hfill \\ November,11\hfill & 59\hfill \\ December,12\hfill & 52\hfill \end{array}$

Calculation:

Here, we know that:

  $\begin{array}{l}Amplitude,A=\frac{LargestValue-SmallestValue}{2}\\ Amplitude,A=\frac{90-51}{2}=19.5\end{array}$

Now, vertical shift will be:

  $\begin{array}{l}verticalshift,B=\frac{LargestValue+SmallestValue}{2}\\ verticalshift,B=\frac{90+51}{2}=70.5\end{array}$

Thus, the data will repeat after $12$ months:

  $T=\frac{2\pi }{12}=\frac{\pi }{6}$

Now, here we will divide $12$ into $4$ parts:

  $\left(0,3\right),\left(3,6\right),\left(6,9\right),\left(9,12\right)$

We can observe that the local maxima occur at $x=3$ , but in the data it is given that the maxima occur at $x=7$ .

Therefore, shift will be:

  $7-3=4$

Now, we will substitute the values in the function:

  $\begin{array}{l}y=A\sin \left(\omega x-\varphi \right)+B\\ y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5\\ y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5\end{array}$

Hence, the required answer is: $y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5$

c.

To determine

Draw the sinusoidal function $y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5$ .

Answer to Problem 95RE

  

Hence, the result is shown in the above diagram.

Explanation of Solution

Given information:

The following data represent average monthly temperatures for phoenix, Arizona.

  $\begin{array}{ll}Month,m\hfill & MonthlyAverageTemperarure,t\hfill \\ January,1\hfill & 51\hfill \\ February,2\hfill & 55\hfill \\ March,3\hfill & 63\hfill \\ April,4\hfill & 67\hfill \\ May,5\hfill & 77\hfill \\ June,6\hfill & 86\hfill \\ July,7\hfill & 90\hfill \\ August,8\hfill & 90\hfill \\ September,9\hfill & 84\hfill \\ October,10\hfill & 71\hfill \\ November,11\hfill & 59\hfill \\ December,12\hfill & 52\hfill \end{array}$

Calculation:

Here, we have:

  

Hence, the result is shown in the above diagram.

d.

To determine

Find the sinusoidal function of best fit the data .

Answer to Problem 95RE

  $y=19.52\sin \left(0.5408x+4\right)+71.01$

Explanation of Solution

Given information:

The following data represent average monthly temperatures for phoenix, Arizona.

  $\begin{array}{ll}Month,m\hfill & MonthlyAverageTemperarure,t\hfill \\ January,1\hfill & 51\hfill \\ February,2\hfill & 55\hfill \\ March,3\hfill & 63\hfill \\ April,4\hfill & 67\hfill \\ May,5\hfill & 77\hfill \\ June,6\hfill & 86\hfill \\ July,7\hfill & 90\hfill \\ August,8\hfill & 90\hfill \\ September,9\hfill & 84\hfill \\ October,10\hfill & 71\hfill \\ November,11\hfill & 59\hfill \\ December,12\hfill & 52\hfill \end{array}$

Calculation:

Here, a curve is fitted using graphic utility.

Thus,

  $y=19.52\sin \left(0.5408x+4\right)+71.01$

Hence, the required answer is. $y=19.52\sin \left(0.5408x+4\right)+71.01$ .

e.

To determine

Graph the sinusoidal function of best fit on the scatter diagram.

Answer to Problem 95RE

  

Explanation of Solution

Given information:

The following data represent average monthly temperatures for phoenix, Arizona.

  $\begin{array}{ll}Month,m\hfill & MonthlyAverageTemperarure,t\hfill \\ January,1\hfill & 51\hfill \\ February,2\hfill & 55\hfill \\ March,3\hfill & 63\hfill \\ April,4\hfill & 67\hfill \\ May,5\hfill & 77\hfill \\ June,6\hfill & 86\hfill \\ July,7\hfill & 90\hfill \\ August,8\hfill & 90\hfill \\ September,9\hfill & 84\hfill \\ October,10\hfill & 71\hfill \\ November,11\hfill & 59\hfill \\ December,12\hfill & 52\hfill \end{array}$

Calculation:

Here, we have function that is best fit on scatter plot, $y=19.5\sin \left(\frac{\pi }{6}\left(x-4\right)\right)+70.5$

  

Hence, the result is shown in the above graph.