945 9.6 Systems of Inequalities and Linear Programming Figure 23(a) shows the region of feasible solutions, together with these lines. The lines are parallel, and the higher the line, the greater the profit. The line 30x + 70y = 7000 yields the greatest profit but does not contain any points of the region of feasible solutions. To find the feasible solution of greatest profit, lower the line 30x + 70y = 7000 until it contains a feasible solution—that is, until it just touches the region of feasible solutions. This occurs at point A, a vertex of the region. The desired maximum value is 5700. See Figure 23(b). Figure 23 x y A 0 10 10 30x + 70y = 7000 30x + 70y = 3000 30x + 70y = 1000 30x + 70y = 0 (a) x y 0 10 10 A 30x + 70y = 7000 30x + 70y = 6300 30x + 70y = 5700 (b) EXAMPLE 5 Minimizing Cost Robin takes multivitamins each day. She wants at least 16 units of vitamin A, at least 5 units of vitamin B1, and at least 20 units of vitamin C. Capsules, costing $0.10 each, contain 8 units of A, 1 of B1, and 2 of C. Chewable tablets, costing $0.20 each, contain 2 units of A, 1 of B1, and 7 of C. How many of each should she take each day to minimize her cost and yet fulfill her daily requirements? What is the minimum cost? SOLUTION Step 1 Let x represent the number of capsules to take each day, and let y represent the number of chewable tablets to take. Then the cost in pennies per day is cost = 10x + 20y. Objective function Robin takes x of the $0.10 capsules and y of the $0.20 chewable tablets, and she gets 8 units of vitamin A from each capsule and 2 units of vitamin A from each tablet. Altogether she gets 8x + 2y units of A per day. She wants at least 16 units, which gives the following inequality for A. 8x + 2y Ú 16 Each capsule and each tablet supplies 1 unit of vitamin B1. Robin wants at least 5 units per day, so the inequality for B is x + y Ú 5. For vitamin C, the inequality is 2x + 7y Ú 20. Because Robin cannot take negative numbers of multivitamins, x Ú 0 and y Ú 0.
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