This problem is a typical linear programming (optimization) scenario where the goal is to maximize the audience reached under budget and other constraints.

Variables:

Let:

• $x_1$ = number of TV ads,
• $x_2$ = number of social media ads,
• $x_3$ = number of billboard ads,
• $x_4$ = number of newspaper ads.

Objective Function:

The objective is to maximize the total audience reached, which is given by: $\text{Maximize } 30,000x_1 + 22,000x_2 + 24,000x_3 + 8,000x_4$ subject to the constraints.

Constraints:

1. Budget constraint: The total cost of all ads cannot exceed $15,000. $800x_1 + 400x_2 + 500x_3 + 100x_4 \leq 15,000$

2. TV/Social media constraint: There should be at least 6 ads from TV or social media combined. $x_1 + x_2 \geq 6$

3. Billboard and newspaper cost constraint: The amount spent on billboards and newspapers together must not exceed the amount spent on TV ads. $500x_3 + 100x_4 \leq 800x_1$

4. Maximum number of ads constraint: Each type of ad has a maximum number allowed, as given in the table: $0 \leq x_1 \leq 10, \quad 0 \leq x_2 \leq 10, \quad 0 \leq x_3 \leq 10, \quad 0 \leq x_4 \leq 10$

Solving Method:

This can be solved using linear programming techniques, such as the Simplex Method or graphical methods if approached manually. Given these constraints, we aim to find the values of $x_1, x_2, x_3, x_4$ that maximize the objective function while satisfying the constraints.

Would you like a detailed step-by-step solution for this problem or further assistance in setting up the solution algorithm?

Here are 5 related questions you could explore:

1. How would the solution change if the budget were increased to $20,000?
2. What if the cost of social media ads increased to $600 per ad?
3. How would the solution differ if there were no limit on the number of newspaper ads?
4. What is the minimum budget required to purchase 6 ads on TV and 6 on social media?
5. How does the audience reached change if the audience per ad is reduced by 10%?

Tip: In linear programming, simplifying the objective function and constraints can often reveal shortcuts or alternative approaches to solving the problem faster.