Chapter 12.3, Problem 74E

a.

To determine

To find: a quadratic model for the given data using the regression feature of a graphing utility.

Answer to Problem 74E

Quadratic model is $y=1.08571x^2-83.9257x+1643.47$ .

Explanation of Solution

Given information: The table shows the number N (in thousands) of books sold when the price per book is p (in dollars).

Price p	Number of books , N
$15	630
$20	396
$25	227
$30	102
$35	36

Calculation:

Using the graphing utility on above table data to find the model:

  $\begin{array}{c}\text{QuadReg}\\ y=a{x}^2+bx+c.\\ a=\mathrm{1.08571.}\\ b=-\mathrm{83.9257.}\\ c=164347.\end{array}$

  $y=1.08571x^2-83.9257x+\mathrm{1643.47.}$

b.

To determine

To graph: the model found in part (a) using graphing utility and estimate the slope of the graph when p =$20 and p =$30.

Answer to Problem 74E

  $\begin{array}{c}\text{slope}x=20,f\text{'}(20)=-\mathrm{40.4973.}\\ \text{slope}x=30,f\text{'}(30)=-\mathrm{18.7831.}\end{array}$

Explanation of Solution

Given information: The table shows the number N (in thousands) of books sold when the price per book is p (in dollars).

Price p	Number of books , N
$15	630
$20	396
$25	227
$30	102
$35	36

Calculation:

Quadratic model is $y=1.08571x^2-83.9257x+1643.47$

The graph of this model using graphing utility is shown below

  

The function is ,

  $\begin{array}{c}f(x)=1.08571x^2-83.9257x+1643.47\\ \text{slope}\text{of}\text{the}\text{function}f\text{'}(x)=2\times 1.08571x-\mathrm{83.9257.}\\ \text{slope}x=20,f\text{'}(20)=-\mathrm{40.4973.}\\ \text{slope}x=30,f\text{'}(30)=-\mathrm{18.7831.}\end{array}$

c.

To determine

To graph: the tangent line to the model when p =$20 and p =$30 using graphing utility. Compare the slope given by the graphing utility with the estimate in part (b).

Answer to Problem 74E

The slop of the tangent found from graph is same as the slope estimated.

Explanation of Solution

Given information: The table shows the number N (in thousands) of books sold when the price per book is p (in dollars).

Price p	Number of books , N
$15	630
$20	396
$25	227
$30	102
$35	36

Calculation:

The tangent line graph to the model when p =20 using graphing utility is shown below.

  

The tangent line graph to the model when p =30 using graphing utility is shown below.

  

The slop of the tangent found from graph is same as the slope estimated.

d.

To determine

To explain: the slope of the tangent lines at p =$20 and p =$30 are not the same what this means to the company selling the books.

Answer to Problem 74E

The slope at x =20 is not equal to the slop at x =30. Slope represents the rate of change of number of books when price is changed. Thus the price change is more affected when price is $20 than $30.

Explanation of Solution

Given information: The table shows the number N (in thousands) of books sold when the price per book is p (in dollars).

Price p	Number of books , N
$15	630
$20	396
$25	227
$30	102
$35	36

Calculation:

  $\begin{array}{c}\text{slope}x=20,f\text{'}(20)=-\mathrm{40.4973.}\\ \text{slope}x=30,f\text{'}(30)=-\mathrm{18.7831.}\end{array}$

The slope at x =20 is not equal to the slop at x =30. Slope represents the rate of change of number of books when price is changed. Thus the price change is more affected when price is $20 than $30.