943 9.6 Systems of Inequalities and Linear Programming Fundamental Theorem of Linear Programming If an optimal value for a linear programming problem exists, then it occurs at a vertex of the region of feasible solutions. EXAMPLE 4 Maximizing Rescue Efforts Earthquake victims in China need medical supplies and bottled water. Each medical kit measures 1 ft3 and weighs 10 lb. Each container of water is also 1 ft3 and weighs 20 lb. The plane can carry only 80,000 lb with a total volume of 6000 ft3. Each medical kit will aid 6 people, and each container of water will serve 10 people. How many of each should be sent to maximize the number of people aided? What is this maximum number of people? SOLUTION Step 1 We translate the statements of the problem into symbols as follows. Let x = the number of medical kits to be sent, and y = the number of containers of water to be sent. Negative values of x and y are not valid for this problem. So x Ú 0 and y Ú 0. Each medical kit and each container of water will occupy 1 ft3 of space, and there is a maximum of 6000 ft3 available. 1x + 1y … 6000 x + y … 6000 Each medical kit weighs 10 lb, and each water container weighs 20 lb. The total weight cannot exceed 80,000 lb. 10x + 20y … 80,000 x + 2y … 8000 Divide by 10. The four inequalities in color form a system of linear inequalities. x Ú 0 y Ú 0 x + y … 6000 x + 2y … 8000 These are the constraints on the variables in this application. Because each medical kit will aid 6 people and each container of water will serve 10 people, the total number of people served is represented by the following objective function. Number of people served = 6x + 10y Multiply the number of items by the number of people served and add. Step 2 The maximum number of people served, subject to these constraints, is found by sketching the graph of the solution set of the system. See Figure 21 on the next page. The only feasible values of x and y are those that satisfy all constraints. These values correspond to points that lie on the boundary or in the shaded region, which is the region of feasible solutions.
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