Questions: A store has a 9000 monthly advertising budget. Newspaper ads cost 600 each with a limit of 40 per month, radio ads cost 30 each with a limit of 60 per month, and TV ads cost 3000 each with a limit of 10 per month. About 5000 people see each newspaper ad, 3000 hear each radio ad, and 12,000 see each TV ad. How much of each type should be used to maximize the store's exposure? Interpret the results. If the store wants to maximize exposure, it should use newspaper ads, radio ads, and TV ads. A marketing analyst is puzzled by these results. More people see each TV ad than each newspaper or radio ad, so it makes no sense to use ads in this proportion. How would you respond? A. Agree, and state that a mistake must have been made in the analysis. B. Disagree, and point out that the TV ads have the lowest limit per month of all the advertising mediums. C. Disagree, and point out that TV ads actually reach the fewest people per dollar. D. None of the above.

A store has a 9000 monthly advertising budget. Newspaper ads cost 600 each with a limit of 40 per month, radio ads cost 30 each with a limit of 60 per month, and TV ads cost 3000 each with a limit of 10 per month. About 5000 people see each newspaper ad, 3000 hear each radio ad, and 12,000 see each TV ad. How much of each type should be used to maximize the store's exposure? Interpret the results. If the store wants to maximize exposure, it should use newspaper ads, radio ads, and TV ads. A marketing analyst is puzzled by these results. More people see each TV ad than each newspaper or radio ad, so it makes no sense to use ads in this proportion. How would you respond? A. Agree, and state that a mistake must have been made in the analysis. B. Disagree, and point out that the TV ads have the lowest limit per month of all the advertising mediums. C. Disagree, and point out that TV ads actually reach the fewest people per dollar. D. None of the above.
Transcript text: A store has a $\$ 9000$ monthly advertising budget. Newspaper ads cost $\$ 600$ each with a limit of 40 per month, radio ads cost $\$ 30$ each with a limit of 60 per month, and TV ads cost $\$ 3000$ each with a limit of 10 per month. About 5000 people see each newspaper ad, 3000 hear each radio ad, and 12,000 see each TV ad. How much of each type should be used to maximize the store's exposure? Interpret the results. If the store wants to maximize exposure, it should use $\square$ newspaper ads, $\square$ radio ads, and $\square$ TV ads. A marketing analyst is puzzled by these results. More people see each TV ad than each newspaper or radio ad, so it makes no sense to use ads in this proportion. How would you respond? A. Agree, and state that a mistake must have been made in the analysis. B. Disagree, and point out that the TV ads have the lowest limit per month of all the advertising mediums. C. Disagree, and point out that TV ads actually reach the fewest people per dollar. D. None of the above.
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Solution

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Solution Steps

Step 1: Define the Variables

Let \( x_1 \) be the number of newspaper ads, \( x_2 \) be the number of radio ads, and \( x_3 \) be the number of TV ads. The objective is to maximize the total exposure given by the equation:

\[ \text{Exposure} = 5000x_1 + 3000x_2 + 12000x_3 \]

Step 2: Set Up the Constraints

The constraints based on the budget and limits are as follows:

  1. Budget constraint: \[ {600x_1 + 30x_2 + 3000x_3 \leq 9000} \]
  2. Newspaper ad limit: \[ x_1 \leq 40 \]
  3. Radio ad limit: \[ x_2 \leq 60 \]
  4. TV ad limit: \[ x_3 \leq 10 \]
  5. Non-negativity constraints: \[ x_1, x_2, x_3 \geq 0 \]
Step 3: Solve the Linear Programming Problem

After solving the linear programming problem, we find the optimal values for the variables:

\[ x_1 \approx 40, \quad x_2 \approx 0, \quad x_3 \approx 0 \]

Final Answer

To maximize exposure, the store should use approximately \( \boxed{40} \) newspaper ads, \( \boxed{0} \) radio ads, and \( \boxed{0} \) TV ads.

Regarding the marketing analyst's confusion, the correct response is C: Disagree, and point out that TV ads actually reach the fewest people per dollar.

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