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Problem 387

# Solve the following linear programming problem: Maximize $$\quad 6 \mathrm{~L}_{1}+11 \mathrm{~L}_{2}$$ subject to: $$\quad 2 \mathrm{~L}_{1}+\mathrm{L}_{2} \leq 104$$ $$\mathrm{L}_{1}+2 \mathrm{~L}_{2} \leq 76$$ and $$L_{1} \geq 0, L_{2} \geq 0$$

Expert verified
The solution of the given linear programming problem is $$L_1 = 20$$ and $$L_2 = 28$$, yielding a maximum objective function value of 504.
See the step by step solution

## Step 1: Graph the Constraints

To graph the constraints, first transform each inequality into an equation: $$2L_1+L_2 = 104$$ and $$L_1 + 2L_2 = 76$$. Now, plot these two lines on the Cartesian coordinate system together with the conditions $$L_1\geq 0$$ and $$L_2\geq 0$$. You should get a feasible region (polygon) enclosed by these lines and the axes.

## Step 2: Find the Vertices of the Feasible Region

Next, find the vertices of the feasible region by finding the points where the constraint lines intersect each other and the axes. 1. Intersection with the x-axis: Solve $$2L_1+L_2 = 104$$ with $$L_2 = 0$$. 2. Intersection with the y-axis: Solve $$L_1+2L_2 = 76$$ with $$L_1 = 0$$. 3. Intersection between constraint lines: Solve the system of equations $$2L_1+L_2 = 104$$ and $$L_1+2L_2 = 76$$. Using these methods, we obtain the following vertices of the feasible region: - Vertex A: $$(0,0)$$ - Vertex B: $$(0,38)$$ - Vertex C: $$(52,0)$$ - Vertex D: $$(20,28)$$

## Step 3: Evaluate the Objective Function at Each Vertex

Now that we have found the vertices of the feasible region, we need to evaluate the objective function at each vertex: 1. Vertex A: (0,0) - Objective Function Value: $$6(0)+11(0)=0$$ 2. Vertex B: (0,38) - Objective Function Value: $$6(0)+11(38)=418$$ 3. Vertex C: (52,0) - Objective Function Value: $$6(52)+11(0)=312$$ 4. Vertex D: (20,28) - Objective Function Value: $$6(20)+11(28)=504$$

## Step 4: Determine the Maximum Value

Now that we have evaluated the objective function at each vertex, we can compare the values to determine which vertex yields the maximum value. The results are: - Vertex A: 0 - Vertex B: 418 - Vertex C: 312 - Vertex D: 504 The maximum value, 504, is achieved at the vertex D with coordinates (20, 28). Therefore, the solution of the given linear programming problem is $$L_1 = 20$$ and $$L_2 = 28$$, yielding a maximum objective function value of 504.

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