Chapter 3.2, Problem 22AYU

a.

To determine

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

Answer to Problem 22AYU

It is not a function.

Explanation of Solution

Given information:

A marketing firm wishes to find a function that relates the sales $S$ of a product and $A$ the amount spent on advertising the product. The data are formed by past experience,advertising and sales are measured in thousands of dollars.

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

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

Calculation:

Consider the table,

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

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 $A=24,S=343and341$

Hence it is not a function.

b.

To determine

Draw a scatter diagram of the data.

Answer to Problem 22AYU

  

Explanation of Solution

Given information:

A marketing firm wishes to find a function that relates the sales $S$ of a product and $A$ the amount spent on advertising the product. The data are formed by past experience,advertising and sales are measured in thousands of dollars.

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

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 22AYU

  $\boxed{D=-1.3355p+86.1974}$

Explanation of Solution

Given information:

A marketing firm wishes to find a function that relates the sales $S$ of a product and $A$ the amount spent on advertising the product. The data are formed by past experience, advertising and sales are measured in thousands of dollars.

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

Use a graphing utility to find the line of best fit that models the relation between advertising and sales.

Calculation:

Using a graphing utility to find the line of best fit that models the relation between advertising and sales,

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

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

d.

To determine

Interpret the slope.

Answer to Problem 22AYU

  $\boxed{-1.3355}$

Explanation of Solution

Given information:

A marketing firm wishes to find a function that relates the sales $S$ of a product and $A$ the amount spent on advertising the product. The data are formed by past experience; advertising and sales are measured in thousands of dollars.

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

Interpret the slope.

Calculation:

The slope of the line is $m=\frac{y_2-y_1}{x_2-x_1}$ where $\left(x_1,y_1\right),\left(x_2,y_2\right)$ two points on the line, are $\left(20,335\right)and\left(24,341\right)$

  $\begin{array}{l}m=\frac{341-335}{24-20}\\ m=\frac{6}{4}\\ m=1.5\end{array}$

Hence, slope of line is $\boxed{m=1.5}$

f.

To determine

What is the domain of the function?

Answer to Problem 22AYU

  $\left\{20,22,22.5,24,27,28.3\right\}.$

Explanation of Solution

Given information:

A marketing firm wishes to find a function that relates the sales $S$ of a product and $A$ the amount spent on advertising the product. The data are formed by past experience, advertising and sales are measured in thousands of dollars.

  $\begin{array}{cccccccc}\begin{array}{l}Adverti\sin g\\ Expenditure,A\end{array}& 20& 22& 22.5& 24& 24& 27& 28.3\\ Sales,S& 335& 339& 338& 343& 341& 350& 351\end{array}$

What is the domain of the function?

Calculation:

The domain of $f$ is the largest set of real numbers for which the value of $f\left(x\right)$ is a real number.

That is the domain of the function consists of the set of $x-value$ .

Thus, the domain of the given relation is $\left\{20,22,22.5,24,27,28.3\right\}.$

Hence, domain of the function is $\left\{20,22,22.5,24,27,28.3\right\}.$