Chapter 11.3, Problem 68E

a.

To determine

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

Answer to Problem 68E

  $f(p)=1.04p^2-81.5p+1613.$

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 (in dollars)	Number of books, n (in thousands)
10	900
15	630
20	396
25	227
30	102
35	36

Calculation:

Using the regression feature of a graphing utility, the quadratic model for the above table data is:

  $f(p)=1.04p^2-81.5p+1613.$

b.

To determine

To graph: the model found in part (a) using the graphing utility and estimate the slope of the graph when p = 15 and p = 30.

Answer to Problem 68E

The slope of the graph when:

  $\begin{array}{l}\text{For}p=15\\ {\text{slope}}_{\tan }\approx -55\\ \text{For}p=30\\ {\text{slope}}_{\tan }\approx -16\end{array}$

Explanation of Solution

Given information: $f(p)=1.04p^2-81.5p+1613.$

Calculation:

The graph of the model $f(p)=1.04p^2-81.5p+1613$ , is given below.

  

  

From the graph

  $\begin{array}{l}\text{For}p=15\\ {\text{slope}}_{\tan }=\frac{\Delta y}{\Delta x}\approx \frac{575-630}{16-15}=-55\\ \text{For}p=30\\ {\text{slope}}_{\tan }=\frac{\Delta y}{\Delta x}\approx \frac{86-102}{31-30}=-16\end{array}$

c.

To determine

To graph: the tangent lines to the model when p = 15 and p = 30. And compare the slope given by the graphing utility with estimates in part (b)

Answer to Problem 68E

f’ (15) = -50.3 and f’ (30) = -19.1.

Explanation of Solution

Given information:

Calculation:

Using a graphing calculator to graph the tangent lines to the model at p = 15 and p = 30, get f’ (15) = -50.3 and f’ (30) = -19.1, which are both fairly close to our approximations.

  

d.

To determine

To explain: the slope of the tangent lines at p = 15 and p = 30 are not the same, what this means to the company selling the books.

Explanation of Solution

Given information:

Calculation:

The slope of the tangent line at p = 15 has a far more negative slope than that at p = 30. This means that the company will see a more radical decline in number of books sold as they increase the price past $15 per book, but that the decline in number of books sold will be more subtle if they increase the price from $30 to a greater amount per book.