Final answer:
To minimize the cost function c = 9x + 9y with given constraints, we graph the constraints, find the feasible region, and evaluate the cost function at each vertex. The minimum cost appears to be at the point (5, 7.5) with c = 112.5. However, this value is not among the provided options, suggesting there might be an error in the question or options.
Step-by-step explanation:
The objective is to minimize the cost function c = 9x + 9y given the constraints:
- x + 2y ≥ 20
- 2x + y ≥ 20
- x ≥ 0
- y ≥ 0
To solve this linear programming problem, we should first graph the constraints to find the feasible region. Then we will identify the vertices of this region and evaluate the cost function at each vertex to find the minimum value.
Graphing the constraints:
- For x + 2y ≥ 20, when x = 0, y = 10 and when y = 0, x = 20.
- For 2x + y ≥ 20, when x = 0, y = 20 and when y = 0, x = 10.
- x ≥ 0 and y ≥ 0 represent the positive quadrant of the coordinate system.
By graphing these lines and finding the feasible region, we can see that the points (0,10), (10,0), and the intersection of the first two lines are the vertices of the feasible region.
Finding the intersection gives us:
- x + 2y = 20
- 2x + y = 20
Solving these two equations simultaneously yields x = 5 and y = 7.5.
Evaluating the cost function at the vertices:
- At (0,10): c = 9(0) + 9(10) = 90
- At (10,0): c = 9(10) + 9(0) = 90
- At (5,7.5): c = 9(5) + 9(7.5) = 45 + 67.5 = 112.5
Therefore, the minimum cost is when x = 5 and y = 7.5, giving c = 112.5, which is not one of the options provided.