372 CHAPTER 6 Algebra, Graphs, and Functions y x (0, 15) 4x 1 8y 5 160 (0, 0) (30, 5) (40, 0) 25 20 15 10 5 5 101520253035 25 4x 1 12y 5 180 Figure 6.38 The list of constraints is a system of linear inequalities in two variables. The solution to the system of inequalities is the set of ordered pairs that satisfy all the constraints. The constraints and the vertices of the feasible region are indicated in Fig. 6.38. Note that the solution to the system consists of the shaded region and the solid boundaries. The vertices at (0, 0), (0, 15), (30, 5), and (40, 0) are the points at which the boundaries intersect. These points can also be determined by the addition or substitution method described in Section 6.7. The goal in this example is to maximize the profit. The objective function is given by the profit formula P x y 40 65 . = + According to the fundamental principle, the maximum profit will be determined at one of the vertices of the feasible region. Calculate P for each one of the vertices. = + = + = = + = = + = = + = P x y P P P P At (0, 0), At (0, 15), At (30, 5), At (40, 0), 40 65 40(0) 65(0) 0 40(0) 65(15) 975 40(30) 65(5) 1525 40(40) 65(0) 1600 The maximum profit is at (40, 0), which means that the company should manufacture 40 plain rocking chairs and no fancy rocking chairs. The maximum profit would be $1600. The minimum profit would be at (0, 0), when no rocking chairs of either style were manufactured. 7 Now try Exercise 51 Example 8 Maximizing Profit The Admiral Appliance Company makes washers and dryers. The company must manufacture at least one washer per day to ship to one of its customers. No more than 6 washers can be manufactured due to production restrictions. The number of dryers manufactured cannot exceed 7 per day. Also, the number of washers manufactured cannot exceed the number of dryers manufactured per day. If the profit on each washer is $20 and the profit on each dryer is $30, how many of each appliance should the company make per day to maximize profits? What is the maximum profit? Solution Let = = = = = x y x y P the number of washers manufactured per day the number of dryers manufactured per day 20 the profit on washers 30 the profit on dryers the total profit The maximum profit is dependent on several constraints. The number of appliances manufactured each day cannot be a negative amount. This condition gives us the constraints x 0 ≥ and y 0. ≥ The company must manufacture at least one washer per day; therefore, x 1. ≥ No more than 6 washers can be manufactured per day; therefore, x 6. ≤ No more than 7 dryers can be manufactured per day; therefore, y 7. ≤ The number of washers cannot exceed the number of dryers manufactured per day; therefore, x y. ≤ Thus, the six constraints are x y x x y x y 0, 0, 1, 6, 7, ≥ ≥ ≥ ≤ ≤ ≤ In this example, the objective function is the profit function. Since x 20 is the profit on x washers and y 30 is the profit on y dryers, the profit function is BearFotos/Shutterstock
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