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Problem 389

# A marketing manager wishes to maximize the number of people exposed to the company's advertising. He may choose television commercials, which reach 20 million people per commercial, or magazine advertising, which reaches 10 million people per advertisement. Magazine advertisements cost $$\ 40,000$$ each while a television advertisement costs $$\ 75,000$$. The manager has a budget of $$\ 2,000,000$$ and must buy at least 20 magazine advertisements. How many units of each type of advertising should be purchased?

Expert verified
The marketing manager should purchase 16 TV commercials and 20 magazine advertisements to maximize exposure within the given budget and constraints. This maximizes the objective function $$Z = 20x + 10y$$, with x = 16 and y = 20.
See the step by step solution

## Step 1: Define variables

Let's use the following variables: - Let x represent the number of TV commercials. - Let y represent the number of magazine advertisements.

## Step 2: Formulate the objective function

The objective is to maximize the number of people exposed to the advertising. Each TV commercial reaches 20 million people and each magazine advertisement reaches 10 million people. We can formulate the objective function like this: Maximize the objective function: $$Z = 20x + 10y$$

## Step 3: Formulate constraint functions

First, let's consider the budget constraint: The manager has a budget of ($$2,000,000$$). Each TV commercial costs $$75,000$$ and each magazine advertisement costs $$40,000$$. So, the constraint is: $$75,000x + 40,000y \leq 2,000,000$$ Second, the manager must buy at least 20 magazine advertisements (constraint for y): $$y \geq 20$$ We also need non-negativity constraints for x and y: $$x \geq 0$$ $$y \geq 0$$

## Step 4: Solve the linear programming problem

We can solve this linear programming problem using the graphical method or any optimization software. By using graphical method or optimization tools, we can find the solution to the problem as follows: x = 16 y = 20

## Step 5: Interpret the solution

The manager should purchase 16 TV commercials and 20 magazine advertisements to maximize the number of people exposed to the company's advertising within the given budget and constraints.

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