Chapter 4.8, Problem 57E

a.

To determine

Create a scatter plot of the data.

Answer to Problem 57E

  

Explanation of Solution

Given information:

The table shows the average sales $S$ (in millions of dollars) of an outerwear manufacturer for each month $t$ , where $t=1$ corresponds to January.

  $\begin{array}{ccccccccccccc}Time,t& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ Sales,S& 13.46& 11.15& 8.00& 4.85& 2.54& 1.70& 2.54& 4.85& 8.00& 11.15& 13.46& 14.3\end{array}$

Create a scatter plot of the data.

Calculation:

Consider the, sales $S$ (in millions of dollars) of an outerwear manufacturer per month.

Now use graphing utility $TI-83$ to plot the graph.

Go to STAT and select EDIT.

  

Enter the data in the line $L_1$ as year and $L_2$ as sales in that list.

  

Now go to STAT PLOT and choose PLOT1.

  

Select the first type,

  

Now, go to zoom and ZOOMSTAT.

  

Put ENTER and get the plot.

  

Hence the result is shown in graph.

b.

To determine

Find a trigonometric model that fits the data.

Answer to Problem 57E

  

  $\boxed{S=6.302\sin \left(0.524x+1.570\right)+8.00}$

Explanation of Solution

Given information:

The table shows the average sales $S$ (in millions of dollars) of an outerwear manufacturer for each month $t$ , where $t=1$ corresponds to January.

  $\begin{array}{ccccccccccccc}Time,t& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ Sales,S& 13.46& 11.15& 8.00& 4.85& 2.54& 1.70& 2.54& 4.85& 8.00& 11.15& 13.46& 14.3\end{array}$

Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data?

Calculation:

Consider the, sales $S$ (in millions of dollars) of an outerwear manufacturer per month.

Now use graphing utility $TI-83$ to plot the graph.

Go to STAT and select CLAC.

  

Then go to the SinReg and go to VARS. And function data in the line $L_1$ as year and $L_2$ as sales in that list.

  

Now go to the ENTER.

  

Hence,the trigonometric model of SALES is,

  $\boxed{S=6.302\sin \left(0.524x+1.570\right)+8.00}$

Now graph the model in same viewing model , now go to $Y=$ option and put the function as blow,

  

Now do not change the window as the WINDOW is adjusted to the scatter plot.

  

Now go to the GRAPH option.

  

Hence, the result shown in graph.

c.

To determine

What is the period of the model? Do you think it is reasonable given the context? Explain.

Answer to Problem 57E

  $\boxed{12months}$

Explanation of Solution

Given information:

The table shows the average sales $S$ (in millions of dollars) of an outerwear manufacturer for each month $t$ , where $t=1$ corresponds to January.

  $\begin{array}{ccccccccccccc}Time,t& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ Sales,S& 13.46& 11.15& 8.00& 4.85& 2.54& 1.70& 2.54& 4.85& 8.00& 11.15& 13.46& 14.3\end{array}$

What is the period of the model? Do you think it is reasonable given the context? Explain.

Calculation:

Find the period of the model consider the plot at $t=0$ .

Now use graphing utility $TI-83$ to plot the graph.

Go to CLAC-I option put $X=0$ and ENTER.

  

  $S\left(0\right)=14.3$

Now go to TRACE option and find the point in which the function has the same value.

  

  $S\left(12\right)=14.3$

Hence, the period of the model is $\boxed{12months}$ .

d.

To determine

Interpret the meaning of the model’s amplitude in the context of the problem.

Answer to Problem 57E

  $\boxed{A=6.302}$

Explanation of Solution

Given information:

The table shows the average sales $S$ (in millions of dollars) of an outerwear manufacturer for each month $t$ , where $t=1$ corresponds to January.

  $\begin{array}{ccccccccccccc}Time,t& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ Sales,S& 13.46& 11.15& 8.00& 4.85& 2.54& 1.70& 2.54& 4.85& 8.00& 11.15& 13.46& 14.3\end{array}$

Interpret the meaning of the model’s amplitude in the context of the problem.

Calculation:

Consider the model,

  $S=6.302\sin \left(0.524x+1.570\right)+8.00$

The wave equation is,

  $\begin{array}{l}y=A\sin \left(\omega t+b\right)+c\\ A=6.302\end{array}$

Hence, the amplitude is $\boxed{A=6.302}$ .