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The problem you're working on seems to cover **linear programming** and **graphical solutions** of inequalities, along with an optimization question about minimizing transportation costs using pick-up trucks and vans. Here's a breakdown and a potential solution outline for each part:

### Part 1
#### a. Plot the feasible region for the system of inequalities:
\[
x \geq 0, \quad y \geq 0, \quad 2x + y \leq 10, \quad x + 2y \leq 8
\]

The goal is to graph these inequalities and find the feasible region where they all overlap. The boundaries can be found by solving the equations:
- \( 2x + y = 10 \) (for which intercepts are \( x = 5 \), \( y = 10 \)).
- \( x + 2y = 8 \) (for which intercepts are \( x = 8 \), \( y = 4 \)).

#### b. Find the maximum and minimum value of the function \( f(x, y) = 5x + 2y \) in the feasible region.
This is a standard linear programming question. You will check the vertices of the feasible region and compute \( f(x, y) \) at those points to find the maximum and minimum.

### Part 2
#### a. Plot the feasible region for the new system of inequalities:
\[
0 \leq x \leq 5, \quad 0 \leq y \leq 5, \quad 2x + 5y \leq 36, \quad 4x + 3y \leq 36
\]
Again, solve the equations of the boundary lines to find their intercepts, and plot them to find the overlap region.

#### b. Maximize the function \( f(x, y) = 2x + 3y \) in the feasible region.
Here, you compute the value of \( f(x, y) \) at the vertices of the feasible region, using linear programming methods.

### Part 3
This involves a factory optimization problem, where you must minimize transportation costs with constraints based on the number of boxes that can fit in vehicles.

#### a. Solve for the feasible region using inequalities:
\[
2x + y \leq 5, \quad x + y \leq 6, \quad x \geq 0, \quad y \geq 0
\]
Find the overlapping feasible region graphically or algebraically.

#### b. Maximize the objective function \( f(x, y) = 50x + 75y \).
After determining the vertices of the feasible region, you substitute them into the objective function to maximize it.

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Would you like help plotting the graphs, solving the inequalities, or computing the maximum and minimum values for each problem? Let me know which part you'd like to focus on!

### Related Questions
1. How do you find the vertices of the feasible region for a system of inequalities?
2. What is the method to determine if the maximum or minimum exists in linear programming?
3. How do you derive the equations for the boundaries of a feasible region?
4. Can you explain how to minimize a cost function using linear programming?
5. What are real-life applications of these types of optimization problems?

### Tip
Always check the points where boundary lines intersect, as these are critical for solving linear programming problems.

Graph the feasible region and optimize the given objective functions based on linear inequalities.

Learn how to solve linear programming problems by graphing the feasible region of inequalities and optimizing objective functions. This problem covers key linear programming concepts and is suitable for students in Grades 11-12.

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