Chapter 3.2, Problem 21AYU

a.

To determine

Does the relation defined by the set of order pairs $\left(p,D\right)$ represent a function?

Answer to Problem 21AYU

It is not a function.

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

Does the relation defined by the set of order pairs $\left(p,D\right)$ represent a function.

Calculation:

Consider the table,

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

No, the relation defined by the set of order pairs $\left(p,D\right)$ represent a function.

Because a function cannot have two values at same input, that is for every $x$ ,there corresponds only one $y$ , although vice versa is not true. As per table data $p=23,D=56and53$

Hence it is not a function.

b.

To determine

Draw a scatter diagram of the data.

Answer to Problem 21AYU

  

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

Draw a scatter diagram of the data.

Calculation:

A scatter diagram of the data of the data is,

  

Hence, the scatter plot is shown in graph.

c.

To determine

Use a graphing utility to find the line of best fit that models the relation between price quantity demanded.

Answer to Problem 21AYU

  $\boxed{D=-1.3355p+86.1974}$

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

Use a graphing utility to find the line of best fit that models the relation between price quantity demanded.

Calculation:

Using a graphing utility to find the line of best fit that models the relation between price quantity demanded,

  $\begin{array}{l}X:vector\left[20,22,23,23,27,29,30\right]\\ Y:vector\left[60,57,56,53,52,49,44\right]\end{array}$

Hence, the line of best fit $\boxed{D=-1.3355p+86.1974}$

d.

To determine

Interpret the slope.

Answer to Problem 21AYU

  $\boxed{-1.3355}$

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

Interpret the slope.

Calculation:

Consider the line, $\boxed{D=-1.3355p+86.1974}$ and compared with linear function of $y=mx+c$

  $m$ is the slope of line,

Hence, slope of line is $\boxed{-1.3355}$

e.

To determine

Express the relationship found in part $\left(c\right)$ using function notation.

Answer to Problem 21AYU

  $\boxed{D\left(p\right)=-1.3355p+86.1974}$

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

Express the relationship found in part $\left(c\right)$ using function notation.

Calculation:

Hence, the relationship found in part $\left(c\right)$ is, $\boxed{D\left(p\right)=-1.3355p+86.1974}$

f.

To determine

What is the domain of the function?

Answer to Problem 21AYU

  $\boxed{\left(0,64.5\right)}$

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

What is the domain of the function?

Calculation:

As we know that price cannot be negative, so the domain of the function will be from $0$ to the point where function touches the horizontal axis, that is, $D=0$ find the value of $p$ .

Put $D=0$ in $D\left(p\right)=-1.3355p+86.1974$ ,

  $\begin{array}{l}0=-1.3355p+86.1974\\ 1.3355p=86.1974\\ p=64.5432\end{array}$

Hence domain of the function is $\boxed{\left(0,64.5\right)}$

g.

To determine

How many jeans will be demanded if the price is $\$28$ ?

Answer to Problem 21AYU

  $\boxed{D\approx 49}$

Explanation of Solution

Given information:

The marketing manager at Levi-Strauss wishes to find a function that relates the demand $D$ for men’s jeans and $p$ , the price of the jeans. The following data were obtained based on a price history of the jeans.

  $\begin{array}{cccccccc}\mathrm{Pr}ice\left(\$/pair\right),p& 20& 22& 23& 23& 27& 29& 30\\ Demand\left(pair,of,jeanssold/Day\right)D& 60& 57& 56& 53& 52& 49& 44\end{array}$

How many jeans will be demanded if the price is $\$28$ ?

Calculation:

Jeans will be demanded if the price is $\$28$ .

  $\begin{array}{l}D=-1.3355\times 28+86.1974\\ D=-37.394+86.1974\\ D=48.8\\ D\approx 49\end{array}$

Hence, Jeans will be demanded are $\boxed{D\approx 49}$