Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

Advertising and Sales Revenue 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

obtained from past experience. Advertising and sales are measured in thousands of dollars.

(a) Does the relation defined by the set of ordered pairs (A, S) represent a function?

(b) Draw a scatter diagram of the data.

(c) Using a graphing utility, find the line of best fit that models the relation between advertising expenditures and sales.

(d) Interpret the slope.

(e) Express the relationship found in part (c) using function notation.

(f) What is the domain of the function?

(g) Predict sales if advertising expenditures are $25,000.

Step-by-Step Solution

Answer

(a). No, the relation defined by the set of ordered pairs (A, S) does not represent a function. 

(b). The scatter diagram is

(c). The line of best fit that models the relation between advertising expenditures and sales is y = 2.0667 x + 292.89.

(d).The slope of the best fit line is 2.0667.

(e).The relationship found in part (c) using function notation is linear and $S\left(A\right) = 2.0667 A + 292.89.$.

(f). The domain of the function is $20\le A\le 28.3$.

(g).The sales if advertising expenditures are

$25,000 is  $51960.39.

Step 1. Given information

The given data table is 

Part (a) Step 1. Explanation

As every distinct value of A is not connected with the distinct value of S.

So, yes the relation defined by the set of ordered pairs (A, S) represent a function 

Part (b) Step 1. Draw the Scatter plot.

The scatter diagram is 

Part (c) Step 1. Best fit line

The line of best fit that models the relation between advertising expenditures and sales is y = 2.0667 x + 292.89.

Part (d) Step 1. Find the slope.

The slope of the best-fit line y = 2.0667 x + 292.89 is 2.0667.  Thus if the advertising expenditures increase by 1 then sales increase by 2.0667. 

Part (e) Step 1. Explanation

The relationship found in part is $S\left(A\right) = 2.0667 A + 292.89$. 

Part (f) Step 1. Find the domain.

The domain of the given function is $20\le A\le 28.3$.

Part (g) Step 1. Sales

Now, if advertising expenditures are $25,000 then the sale is,

$\begin{array}{rcl}y & =& 2.0667\left(25000\right) + 292.89\\ & =& \$51960.39\end{array}$