Chapter 12.3, Problem 74E

a.

To determine

Find a quadratic model for the data.

Answer to Problem 74E

  $\boxed{y=1.0857x^2-83.9257x+1643.4857}$

Explanation of Solution

Given information:

The data in the table show the number $N$ (in thousands) of books sold when the price per book is1y1y3 $p$ (in dollars).

Use the regression feature of a graphing utility to find a quadratic model for the data.

Calculation:

Using as number of years since 2000, the given table can be replaced as

  $\begin{array}{cc} \text{price }p& \text{number }N\\ 15& 630\\ 20& 396\\ 25& 227\\ 30& 102\\ 35& 36\end{array}$

Enter the data from given table into your graphing utility calculator.

To enter the data first press

  $STAT$ the key and then press $1$ to select $Edit$ .

  

  

Now input all the data given sets followed by $ENTER$ key.

  

Before you use regression feature, make sure to enter the DignosticOn command stored in $CATALOG$ menu.

The calculator does not show the value of $r$ regression coefficient unless the DignosticOn command has been entered.

  

  

Press the $STAT$ key and then switch to $CALC$ menu.

Move the cursor to the quadratic regression function.

  

  

Press $ENTER$ key or $6$ key to select the option $6$ : QuadReg and then write the argument $Y_1$ .

Regression equation is stored in $Y_1$ for further use.

  

Now press $ENTER$ key to find the Quadratic Regression equation.

  

Hence, from the above display, write the Quadratic Regression equation $y\approx a{x}^2+bx+c$ as,

  $\boxed{y=1.0857x^2-83.9257x+1643.4857}$ .

b.

To determine

Estimate the slope of the graph.

Answer to Problem 74E

  $\boxed{-40.5}$

  $\boxed{-18.8}$

Explanation of Solution

Given information:

The data in the table show the number $N$ (in thousands) of books sold when the price per book is $p$ (in dollars).

Use the graphing utility to graph the model found in part (a). Estimate the slope of the graph when $p=\$20$ and $p=\$30$ .

Calculation:

The regression equation $y=1.0857x^2-83.9257x+1643.4857$ is already stored in $Y_1$ .

Press the $WINDOW$ key and then set up the viewing window as $[15,35,X_{scl}=5]$ by $[0,700,Y_{scl}=100]$ :

  

Press the $GRAPH$ key to obtain the graph of $Y_1$ in viewing window $[15,35]$ by $[0,700]$ as shown below:

  

To estimate slope of $Y_1$ at $x=20$ using graphic calculator, input the command $nDeriv\left(Y_1,X,20\right)$

  

To copy the function $nDeriv$ ,press the $MATH$ key and then scroll down to select the option $8$ : $nDeriv$ .

To estimate slope of $Y_1$ at $x=30$ using graphic calculator, input the command $nDeriv\left(Y_1,X,30\right)$

  

To copy the function $nDeriv$ ,press the $MATH$ key and then scroll down to select the option $8$ : $nDeriv$ .

Hence, from the above display, slope of the graph when $p=\$20$ is about $\boxed{-40.5}$ and slope of the graph when $p=\$30$ is about $\boxed{-18.8}$ .

c.

To determine

Graph the tangent line to the model.

Compare the slope.

Answer to Problem 74E

  

  

  $\boxed{same}$ .

Explanation of Solution

Given information:

The data in the table show the number $N$ (in thousands) of books sold when the price per book is $p$ (in dollars).

Use the graphing utility to graph the tangent lines to the model $p=\$20$ and $p=\$30$ . Compare the slopes given by the graphing utility with your estimates in part (b).

Calculation:

To graph the tangent line to the $Y_1$ at $x=20$ ,press the key and then select the option $5$ : Tangent.

  

  

Input the value $x=20$ and then press the $ENTER$ key to draw the tangent line. The graphing calculator also gives the equation of tangent line in the slope-intercept form $y=mx+b$ , where coefficient of $x$ is slope $m$ .

  

From the above display, at $x=20$ the coefficient of $x$ is about $-40.5$ , which is slope of the function $Y_1$ .

Hence, the slope obtained above $-40.5$ is $same$ as estimated slope in part b.

To graph the tangent line to the $Y_1$ at $x=30$ ,press the key and then select the option $5$ : Tangent.

Input the value $x=30$ and then press the $ENTER$ key to draw the tangent line.

  

From the above display, at $x=30$ the coefficient of $x$ is about $-18.8$ , which is slope of the function $Y_1$ .

Hence, the slope obtained above $-40.5$ and the slope obtained above $-18.8$ is $same$ as estimated slope in part b.

d.

To determine

Explain what this means to the company selling the books.

Answer to Problem 74E

If the price per book is different,the then the number of books sold will be different.

Explanation of Solution

Given information:

The data in the table show the number $N$ (in thousands) of books sold when the price per book is $p$ (in dollars).

The slopes of the of the tangent at $p=\$20$ and $p=\$30$ are not same. Explain what this means to the company selling the books.

Calculation:

The slope of the tangent at $p=\$20$ is about $-40.5$ and $p=\$30$ is about $-18.8$ .

The different slopes of line represents that if the price per book is different,the then the number of books sold will be different.