To model sales of its tires, the manufacturer of GRIPPER tires used the quadratic equation , where S is regional sales in millions of dollars, x is TV advertising expenditures in millions of dollars, and y is other promotional expenditures in millions of dollars. (See the Extended Application/Group Project “Marginal Return to Sales,” in Chapter 9.)
Although this model represents the relationship between advertising and sales dollars for small changes in advertising expenditures, it is clear to the vice president of advertising that it does not apply to large expenditures for TV advertising on a national level. He knows from experience that increased expenditures for TV advertising result in more sales, but at a decreasing rate of return for the product.
The vice president is aware that some advertising agencies model the relationship between advertising and sales by the function
where , S is sales in millions of dollars, x is TV advertising expenditures in millions of dollars, and y is other promotional expenditures in millions of dollars.* The equation
has the form mentioned previously as being used by some advertising agencies. For TV advertising expenditures up to $20 million, this equation closely approximates
which, in the Extended Application/Group Project in Chapter 9, was used with fixed promotional expenses to describe advertising and sales in Region 1.
To help the vice president decide whether this is a better model for large expenditures, answer the following questions.
What is ? Does this indicate that sales might actually decline after some amount is spent on TV advertising? If so, what is this amount?
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