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Linear Programming Problem Solution

Steps for Solving:

Step 1: Identify the Boundary Constraints of the Feasible Region:

Constraint 1: $x + y \leq 250$

Constraint 2: $x \leq 80$

Constraint 3: $y \geq 40$

Non-negativity constraints: $x \geq 0, y \geq 0$

Step 2: Find the Intersection Points:

Determine the points where these constraints intersect to form the feasible region.

Step 3: Evaluate the Objective Function at Each Point:

Calculate the value of the objective function at each of these points to find the maximum.

Step 4: Choose the Point that Gives the Maximum Profit.

Solution:

Constraint 1: $x + y \leq 250$ (equivalent to $x + y = 250$).

Constraint 2: $x \leq 80$ (this is a vertical constraint).

Constraint 3: $y \geq 40$ (this is a horizontal constraint).

Finding the Intersection Points:

Intersection of Constraint 1 and Constraint 2: $80 + y = 250$, hence $y = 170$. Point: $(80, 170)$.

Intersection of Constraint 1 and Constraint 3: $x + 40 = 250$, hence $x = 210$. Point: $(210, 40)$.

Intersection of Constraint 2 and Constraint 3: $x = 80, y = 40$. Point: $(80, 40)$.

Calculating the Objective Function at Each Point:

At $(80, 170)$: $Z = 0.2(80) + 0.1(170) = 16 + 17 = 33$.

At $(210, 40)$: $Z = 0.2(210) + 0.1(40) = 42 + 4 = 46$.

At $(80, 40)$: $Z = 0.2(80) + 0.1(40) = 16 + 4 = 20$.

Conclusion: The point $(210, 40)$ gives the maximum profit of 46. Therefore, invest $210 in $A_1$ and $40 in $A_2$ to achieve the maximum profit of $46.

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Would you like more details or have any questions?

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1. What is the significance of the feasible region in linear programming?
2. How do non-negativity constraints affect the solution?
3. Can the objective function have multiple maximum values?
4. What happens if no feasible region exists?
5. How would the solution change if the constraints were altered?

Tip: Always check the feasibility of the intersection points within the context of all constraints to ensure a valid solution.