Problem 57 - College Algebra

Question

Begin by deciding on a product that interests the group because you are now in charge of advertising this product. Members were told that the demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. However, as more money is spent on advertising, the price of your product rises. Under what conditions would members recommend an increased expense in advertising? Once you've determined what your product is, write formulas for the given conditions and experiment with hypothetical numbers. What other factors might you take into consideration in terms of your recommendation? How do these factors affect the demand for your product?

Step-by-Step Solution

Answer

If the increase in demand due to advertising outweighs the decrease in demand due to an increase in price, then an increase in advertising spending is recommended. Other influencing factors to consider include market competition, general economy, and seasonal demand.

Step 1: Understanding the Problem

First, understand the problem. The demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. This means that the more money spent on advertising, the more demand. But, as the product's price increases, the demand decreases.

Step 2: Setting up Equations

Next, set up equations to represent these relationships. Let \(d\) be the demand, \(a\) be the amount spent on advertising, and \(p\) be the price of the product. From the direct relationship, we have \(d = ka\), for some constant \(k\). From the inverse relationship, we have \(d = \frac {m}{p}\), for some constant \(m\).

Step 3: Combining Equations

As the advertising cost affects the price, and both of these affect the demand, combine these two equations to get a comprehensive equation that incorporates all three variables. \(d = \frac {ka}{p}\)

Step 4: Interpreting the Results

This equation tells you that the demand will increase if the advertising cost increases and if the price is low. If the increase in demand due to advertising is greater than the decrease in demand due to the price rise resulting from the advertisement, it makes sense to recommend increasing the advertising expense.

Step 5: Considering Other Factors

Finally, consider other factors. Possible factors could include the general economy, market competition, and seasonal demand. All these factors could influence the constants in your equations and therefore alter your results.