This is an optimization problem that can be formulated as a linear programming problem. We want to maximize the number of potential customers who will view the ads, given constraints on the budget and the total number of ads.

Variables

Let:

• $x_1$ be the number of daytime TV ads.
• $x_2$ be the number of prime-time TV ads.
• $x_3$ be the number of late-night TV ads.

Objective Function

We aim to maximize the number of potential customers reached by the ads. Therefore, the objective function to maximize is: $\text{Maximize: } 14,000x_1 + 24,000x_2 + 18,000x_3$ where:

• $14,000x_1$ represents the number of customers viewing daytime ads,
• $24,000x_2$ represents the number of customers viewing prime-time ads,
• $18,000x_3$ represents the number of customers viewing late-night ads.

Constraints

1. Budget constraint: The store can spend up to $22,000. The costs for each ad are:

   • Daytime TV ads cost $1,000,
   • Prime-time TV ads cost $2,000,
   • Late-night TV ads cost $1,500.

   So, the total cost must satisfy: $1000x_1 + 2000x_2 + 1500x_3 \leq 22,000$

2. Total ads constraint: The station will not accept more than 20 ads in total: $x_1 + x_2 + x_3 \leq 20$

3. Non-negativity constraint: The number of ads cannot be negative: $x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0$

Formulation of the Linear Programming Problem

Maximize: $14,000x_1 + 24,000x_2 + 18,000x_3$ Subject to: $1000x_1 + 2000x_2 + 1500x_3 \leq 22,000$ $x_1 + x_2 + x_3 \leq 20$ $x_1, x_2, x_3 \geq 0$

Solution

This linear programming problem can be solved using methods such as the Simplex Method or a solver, but I can help you explore optimal combinations by testing the constraints and maximizing the objective function.

Let me know if you'd like to solve this algebraically or using a numerical solver for the exact ad placements!

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Do you want further details on this problem setup, or do you want to explore solutions using specific methods?

Here are 5 questions to consider for deeper understanding:

1. How would the result change if the budget increased to $30,000?
2. What happens if more weight is placed on prime-time ads?
3. How would the scenario change if there were a cap on the number of daytime ads?
4. What if the station offered a discount for buying ads in bulk?
5. Could we develop a scenario where the number of ads is not capped?

Tip: Linear programming problems are often visualized using constraints as boundaries in a feasible region, making it easier to see where the optimal solution lies.