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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?

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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.

How many ads of each type should be placed to maximize the total number of people reached, given a budget of $15,000 and constraints on the types and number of ads?

Explore a linear programming optimization problem involving maximizing the total audience reached through TV, social media, billboard, and newspaper ads. This problem includes a $15,000 budget constraint and limitations on the number of ads for each medium, making it ideal for undergraduate-level students.

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