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Section 5-2 Systems of Linear Inequalities Example 1

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Section 5-2 Systems of Linear Inequalities
Example 1
A patient in a hospital is required to have at least 84 units of drug A and 120 units of drug B each day (assume an overdose of either drug is harmless). Each gram of substance M contains 10 units of drug A and 8 units of drug B. Each gram of substance N contains 2 units of drug A and 4 units of drug B. How many grams of substances M and N can be mixed to meet the minimum daily requirements?

 Substance Grams M x1 N x2

 Substance M Substance N Constraint Drug A 10x1 + 2x2 ≥ 84 Drug B 8x1 + 4x2 ≥ 120

 Points on Line for Drug A x1 x2 0 42 8.4 0

 Points on Line for Drug B x1 x2 0 30 15 0

The feasible region is shaded in the graph to the left. The point of intersection is at (4, 22):

or

Suppose overdoses are potentially harmful and the patient is to receive between 84 and 100 units of drug A and between 120 and 130 units of drug B. How does this change the feasible region.

 Substance Grams M x1 N x2

 Substance M Substance N Constraint Drug A 10x1 + 2x2 ≥ 84 Drug A 10x1 + 2x2 ≤ 100 Drug B 8x1 + 4x2 ≥ 120 Drug B 8x1 + 4x2 ≤ 130

 Drug A Line 1 Line 2 x1 x2 x1 x2 0 42 0 50 8.4 0 10 0

 Drug B Line 1 Line 2 x1 x2 x1 x2 0 30 0 32.5 15 0 16.25 0

The feasible region is the dark blue region that lies between the two Drug A constraints (the red lines) and, at the same time, between the two Drug B constraints (the green lines). The vertices of the feasible region are found be finding the intersections of the corresponding lines. Working clockwise from the upper-most vertex:

Example 2
A manufacturing plant makes two types of inflatable boats; a two-person boat and a four-person boat. Each two-person boat requires 0.9 labor-hours in the cutting department and 0.8 labor-hours in the assembly department. Each four-person boat requires 1.8 labor hours in the cutting department and 1.2 labor-hours in the assembly department. The maximum labor-hours available each month in the cutting department is 864 and in the assembly department is 672. How many of each type of boat can be manufactured under these constraints?

 Boat # 2-Person x1 4-Person x2

 2-Person 4-Person Constraint Cutting 0.9x1 + 1.8x2 ≤ 864 Assembly 0.8x1 + 1.2x2 ≤ 672

 Cutting x1 x2 0 480 960 0

 Assembly x1 x2 0 560 840 0

The feasible region is shaded in the graph below. The point of intersection is (480, 240) corresponding to 480 2-person boats and 240 4-person boats:

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