A hardware store will run an advertising campaign using radio and newspaper. Every dollar spent on radio advertising will reach 60 people in the "Under $25,000" market, and 50 people in the "Over $25,000" market. Every dollar spent on newspaper advertising will reach 70 people in the "Under $25,000" market, and 40 people in the "Over $25,000" market. If the store wants to reach at least 586,000 people in the "Under $25,000" market and 600,000 people in the "Over $25,000" market, how much should it spend on each type of advertising to minimize the cost?
To solve this problem, we can set up a linear programming model. Let's define our decision variables:
Let x be the amount of money (in dollars) spent on radio advertising.
Let y be the amount of money (in dollars) spent on newspaper advertising.
We aim to minimize the cost, which is given by the objective function:
Cost = x + y
Now let's define the constraints based on the number of people reached:
For the "Under $25,000" market:
60x + 70y ≥ 586,000
For the "Over $25,000" market:
50x + 40y ≥ 600,000
Since we want to reach at least the specified number of people, these are inequality constraints.
To summarize the linear programming model:
Minimize: Cost = x + y
Subject to:
60x + 70y ≥ 586,000 (Constraint for "Under $25,000" market)
50x + 40y ≥ 600,000 (Constraint for "Over $25,000" market)
x ≥ 0 (Non-negativity constraint)
y ≥ 0 (Non-negativity constraint)
Now we can solve this linear programming problem using various methods, such as the Simplex algorithm or graphical methods. The optimal solution will provide the amounts to spend on radio (x) and newspaper (y) to minimize the cost, while still reaching the target number of people in each market segment.