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You provided two images. Let's break down the tasks from each:

1. **First Image:**
   - **Problem:** You are tasked with setting up an objective function and constraints for a Linear Programming (LP) problem to minimize the total miles traveled by bus for students. Each high school has a capacity of 900 students.
   
2. **Second Image:**
   - **Problem:** This is an advertising budget optimization problem. You need to decide how many ads to place in different types (TV, Social Media, Billboards, Newspapers) to maximize the audience reached, subject to a budget of $15,000. There are constraints on the minimum number of ads (at least 6 on TV or Social Media), and the amount spent on billboards and newspapers must not exceed that spent on TV ads.

Let's first tackle the **first problem** (school bus miles minimization):

### 1. Setting up the Linear Programming (LP) for School Bus Problem

Let:
- $ x_1, x_2, \dots, x_n $ be the number of students traveling from one school to another, and
- $ c_1, c_2, \dots, c_n $ be the distances (in miles) they need to travel by bus.

**Objective Function:**
Minimize the total number of miles traveled:
\[
\text{Minimize} \ Z = c_1x_1 + c_2x_2 + \dots + c_nx_n
\]
Where $ x_i $ represents the number of students traveling on each route, and $ c_i $ is the distance for each route.

**Constraints:**
- The total number of students traveling must not exceed 900 per school:
\[
x_1 + x_2 + \dots + x_n \leq 900
\]
- Additionally, there may be constraints on the number of students traveling per route or capacity limits on buses.

### 2. Setting up the LP for the Advertising Budget Problem

Let:
- $ x_1 $ be the number of TV ads,
- $ x_2 $ be the number of Social Media ads,
- $ x_3 $ be the number of Billboard ads, and
- $ x_4 $ be the number of Newspaper ads.

**Objective Function:**
Maximize the total audience reached:
\[
\text{Maximize} \ Z = 30,000x_1 + 22,000x_2 + 24,000x_3 + 8,000x_4
\]

**Constraints:**
1. Budget constraint: The total cost must not exceed $15,000:
\[
800x_1 + 400x_2 + 500x_3 + 100x_4 \leq 15,000
\]
2. Minimum number of ads for TV or Social Media: There should be at least 6 ads from either TV or Social Media:
\[
x_1 + x_2 \geq 6
\]
3. Billboard and Newspaper ads cost constraint: The amount spent on Billboards and Newspapers must not exceed the amount spent on TV ads:
\[
500x_3 + 100x_4 \leq 800x_1
\]
4. Maximum number of ads: There is a maximum limit of 10 ads for each type:
\[
x_1 \leq 10, \ x_2 \leq 10, \ x_3 \leq 10, \ x_4 \leq 10
\]
5. Non-negativity constraints:
\[
x_1, x_2, x_3, x_4 \geq 0
\]

Would you like me to solve any of these optimization problems, or provide further details? Here are some related questions for exploration:

1. How would the solution change if the capacity of each high school increased to 1,000?
2. What if the transportation cost per mile varied for different bus routes?
3. How would changes in the advertising budget affect the optimal allocation of ads?
4. What if certain schools had priority for minimizing bus miles due to distance or congestion?
5. Can the allocation of billboard or newspaper ads be more cost-effective under different circumstances?

**Tip:** Linear programming is highly sensitive to constraints; changing any limit can lead to completely different solutions.

Set up the objective function and constraints for minimizing student miles traveled by bus, and also maximize the number of people reached by ads subject to cost constraints.

Learn how to set up and solve Linear Programming problems including minimizing bus miles traveled by students and maximizing audience reach within a set advertising budget. Includes detailed constraints and objective functions.

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