Chapter 4.8, Problem 69E

a.

To determine

To create: A scatter plot for the given data.

Explanation of Solution

Given Information: Relevant data values arranged in table.

Graph:

  

b.

To determine

To find: A trigonometric model that fits the data and to graph the model on scatter plot and then to determine how well the model fits the data.

Answer to Problem 69E

The model $S=3.150075\cdot sin(0.524t+1.568389)+3.99931$ fits the data very accurately.

Explanation of Solution

Given Information: A relevant table and scatter plot from previous part.

Calculation:

Scatter plot appeared sinusoidal; using sinusoidal regression feature of graphing calculator,

  $\begin{array}{l}SinReg\\ y=a*sin\left(bx+c\right)+d\\ a=3.150075\\ b=0.524\\ c=1.568389\\ d=3.99931\end{array}$

  $\Rightarrow$Equation of model is $S=3.150075\cdot sin(0.524t+1.568389)+3.99931$ .

  

Points lie on equation’s graph, this implies that model fits the data very accurately.

Thus, the model $S=3.150075\cdot sin(0.524t+1.568389)+3.99931$ fits the data very accurately.

c.

To determine

To find: The period of the model and explain if it is reasonable given the context.

Answer to Problem 69E

Period of $12$ months is reasonable as the outwear sales are seasonal.

Explanation of Solution

Given Information: Data is about outerwear sales.

Calculation:

  $Period=\frac{2\pi }{0.524}=11.9908117$

Period is about $12$ months which is reasonable as the outwear sales are seasonal and the point of peak sales is reached on an year-to-year basis.

Thus, period of $12$ months is reasonable as the outwear sales are seasonal.

d.

To determine

To interpret: The meaning of model’s amplitude in the context of the problem.

Answer to Problem 69E

Amplitude implies that greatest average sales are $\$7.15million$ and least average sales are $\$0.85million$ .

Explanation of Solution

Given Information: Data is about outerwear sales, model from one of the previous parts.

Calculation:

  $a=3.15$, $d=4$

The value of amplitude is about $3.15$ , adding $4$ to it gives the greatest average sales $4+3.15=\$7.15million$ and subtracting it from returns least average sales $4-3.15=\$0.85million$ .

Thus, amplitude implies that greatest average sales are $\$7.15million$ and least average sales are $\$0.85million$ .