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Problem 12

Use the method of this section to solve each linear programming problem. \begin{aligned} \text { Minimize } & C=x+2 y \\ \text { subject to } & 4 x+7 y \leq 70 \\ & 2 x+y=20 \\ & x \geq 0, y \geq 0 \end{aligned}

Expert verified
The solution to the given linear programming problem is to minimize the cost function $$C = x + 2y$$, subject to the constraints $$4x + 7y \leq 70$$, $$2x + y = 20$$, and $$x \geq 0$$, $$y \geq 0$$. By graphing the constraints and finding the vertices of the feasible region, we find four vertices: $$(0, 0)$$, $$(10, 0)$$, $$(8, 4)$$, and $$(0, 10)$$. Evaluating the cost function at each vertex, we find that the minimum value of the cost function is $$C_{min} = 0$$ at $$(x, y) = (0, 0)$$.
See the step by step solution

Step 1: Graph constraints and find feasible region

First, let's graph the constraints: $$4x + 7y \leq 70$$ $$2x + y = 20$$ $$x \geq 0$$ $$y \geq 0$$ After graphing, the feasible region is the area where all constraints are satisfied, bounded by the respective lines.

Step 2: Determine the vertices of the feasible region

Observe the feasible region in the graph. The vertices are the points of intersection of the respective lines. There are a total of 4 vertices: 1. Intersection of $$x=0$$ and $$y=0$$ 2. Intersection of $$y=0$$ and $$2x + y = 20$$ 3. Intersection of $$4x + 7y \leq 70$$ and $$2x + y = 20$$ 4. Intersection of $$x=0$$ and $$4x + 7y \leq 70$$ Now, find the coordinates of these vertices: 1. Vertex 1: $$(0, 0)$$ 2. Vertex 2: Solve the system of equations $$y=0$$ and $$2x + y = 20$$ to get $$(10, 0)$$ 3. Vertex 3: Solve the system of equations $$4x + 7y = 70$$ and $$2x + y = 20$$ to get $$(8, 4)$$ 4. Vertex 4: Solve the system of equations $$x=0$$ and $$4x + 7y = 70$$ to get $$(0, 10)$$

Step 3: Calculate the cost function at each vertex

Now, we will evaluate the cost function $$C = x + 2y$$ for every vertex: 1. Vertex 1: $$C(0, 0) = 0$$ 2. Vertex 2: $$C(10, 0) = 10$$ 3. Vertex 3: $$C(8, 4) = 16$$ 4. Vertex 4: $$C(0, 10) = 20$$

Step 4: Identify the minimum value of the cost function

Finally, we will compare the values of the cost function at each vertex to determine the minimum value: - Vertex 1: $$C = 0$$ - Vertex 2: $$C = 10$$ - Vertex 3: $$C = 16$$ - Vertex 4: $$C = 20$$ As we can see, the minimum value of the cost function is at Vertex 1, with $$C=0$$ being the minimum value. Therefore, the solution to this linear programming problem is $$C_{min} = 0$$, at $$(x, y) = (0, 0)$$.

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