To solve the linear programming problem, we need to determine the corner points of the feasible region defined by the constraints and then evaluate the objective function at each of these corner points. Let's break down the problem step by step.

Given Problem

Objective Function: $\text{Maximize } P = 3x + 2y$

Subject to the constraints: $8x + 5y \leq 40$ $5x + 9y \leq 45$ $x \leq 4$ $y \leq 4$ $x \geq 0, \, y \geq 0$

Step 1: Identify the Constraints

The constraints represent the boundaries of the feasible region. We can graph these inequalities to find the corner points (vertices) of the feasible region.

Step 2: Find the Intersection Points (Corner Points)

1. Intersection of $8x + 5y = 40$ and $5x + 9y = 45$:

   Solve the system of equations: $8x + 5y = 40$ $5x + 9y = 45$ To solve, let's use the substitution or elimination method.

   Multiply the first equation by 9 and the second by 5: $72x + 45y = 360$ $25x + 45y = 225$ Subtract the second equation from the first: $47x = 135 \quad \Rightarrow \quad x = \frac{135}{47} \approx 2.87$ Substitute $x = \frac{135}{47}$ back into one of the original equations: $8\left(\frac{135}{47}\right) + 5y = 40$ Simplifying, $y \approx 2.13$ So, one corner point is approximately $\left(\frac{135}{47}, \frac{200}{47}\right) \approx (2.87, 2.13)$.

2. Intersection of $8x + 5y = 40$ and $x = 4$: $8(4) + 5y = 40 \quad \Rightarrow \quad 32 + 5y = 40 \quad \Rightarrow \quad y = \frac{8}{5} = 1.6$ So, the corner point is $(4, 1.6)$.

3. Intersection of $5x + 9y = 45$ and $y = 4$: $5x + 9(4) = 45 \quad \Rightarrow \quad 5x + 36 = 45 \quad \Rightarrow \quad 5x = 9 \quad \Rightarrow \quad x = \frac{9}{5} = 1.8$ So, the corner point is $(1.8, 4)$.

4. Intersection of $x = 4$ and $y = 4$: This is simply the point $(4, 4)$.

5. Intersection of $x = 0$ and $y = 0$: This is the origin $(0, 0)$.

Step 3: List All Corner Points

The feasible region is bounded by the following corner points:

1. $(0, 0)$
2. $(4, 0)$ [Intersection of $x \leq 4$ and $y = 0$]
3. $(0, 4)$ [Intersection of $y \leq 4$ and $x = 0$]
4. $(4, 1.6)$
5. $(1.8, 4)$
6. $(2.87, 2.13)$

Step 4: Conclusion

There are 6 corner points included in the feasible region.

Would you like to evaluate the objective function at each of these points to determine the maximum value, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How do you determine if a solution is optimal in linear programming?
2. What is the significance of corner points in linear programming?
3. How does the Simplex method solve linear programming problems?
4. What happens if the feasible region is unbounded?
5. How can duality be used to understand linear programming solutions?

Tip: When solving linear programming problems graphically, ensure all intersections are considered, and evaluate the objective function at each corner to find the optimal solution.