Chapter 11.3, Problem 68E

Solution

(a)

To find: the quadratic model for the data.

The quadratic model: $f\left(p\right)=1.04p^2–81.5p+1613$ .

Given:

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

Calculation:

Here, use the quadratic regression function of your calculator, inputting the given data to find a quadratic model for it. Use the price, p as the "x" value, and Number of books, "n" as a y value. You should get the quadratic model:

  $f\left(p\right)=1.04p^2–81.5p+1613$

Conclusion:

Therefore, the quadratic model: $f\left(p\right)=1.04p^2–81.5p+1613$ .

(b)

To estimate: the slopes of the graph ad graph the model

The slope of the graph is $-55$ and $-16$ .

  

Calculation:

Graph the quadratic model you got in part (a). Now use this graph to estimate the slopes of the graph when $p=15$ and $p=30$ by using viewing windows tightly focused on those areas and measuring the apparent rates of change in y and x at those points:

For $p=15$ :

  $\begin{array}{l}Slop{e}_{\tan }=\frac{\Delta y}{\Delta x}\\ \approx \frac{575-630}{16-15}\\ =-55\end{array}$

For $p=30$ :

  $\begin{array}{l}Slop{e}_{\tan }=\frac{\Delta y}{\Delta x}\\ \approx \frac{86-102}{31-30}\\ =-16\end{array}$

  

Conclusion:

Therefore,the slope of the graph is $-55$ and $-16$ .

(c)

To compare: the slopes given by the graphing utility and graph the tangent lines

Slopes are both fairly close to our approximations.

  

Calculation:

Using a graphing calculator to graph the tangent lines to the model at $p=15$ and $p=30$ , it is obtained $f\text{'}\left(15\right)=-50.3$ and $f\text{'}\left(30\right)=-19.1$ , which are both fairly close to our approximations.

  

Conclusion:

Therefore,slopes are both fairly close to our approximations.

(d)

To explain: about the tangent lines which are not same

The decline in number of books sold will be more subtle if they increase the price from $30 to a greater amount per book.

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.

  

Conclusion:

Therefore,the decline in number of books sold will be more subtle if they increase the price from $30 to a greater amount per book.