Problems:
Problem 1:
Graph the following systems of inequalities; find the region and the vertices.
a) 8x+10y≤40
4x-6y ≤-12
b) 4x-3y≤12
5x+2y≤10
x≥0,y≥0
Problem 2:
Solve the following linear programming graphically:
a) Maximize P = 20x+15y
x+3y≤18
support to x+y ≤12
x≥0,y≥0
b) Maximize P = 3x-4y
x+3y≤15
subject to 4x+y≤16
x ≥0,y≥0
Problem 3:
Solve the following linear programming graphically:
a) C = 3x+2y
2x+3y≥90
subject to 3x+2y≥120
x ≥0,y≥0
Problem 4:
Solve by the Simplex Method
a) Maximize P = 20x+7y
x+3y≤20
subject to 5x+2y≤30
x ≥0,y≥0
b) Maximize P = 5x+4y+6z
x+y+2z≤10
x+2y+4z≤28
subject to 3x+4y+3z≤18
x ≥0,y≥0,z≥0
Problem 5:
For each of the two problems, identify the inequalities, the objective function, the feasible set, the possible solutions, and the maximum or minimum.
a) A nutritionist is designing a new breakfast cereal using wheat germ and enriched oat flour as the basic ingredients. Each ounce of wheat germ contains 2 milligrams of niacin, 3 milligrams of iron, and 0.5 milligram of thiamin and costs 3 cents. Each ounce of enriched oat flour contains 3 milligrams of niacin, 3 milligram of iron, and 0.25 milligram of thiamin and cost 4 cents. The nutritionist wants the cereal to have at least 7 milligrams of niacin, 9 milligrams of iron, and 1 milligram of thiamin. How many ounces of wheat germ and how many ounces of enriched oat flour should be used in each serving to meet the nutritional requirements at the least cost?
b) A confectioner makes two raisin-nut mixtures. A box of mixture A contains 6 ounces of peanuts, 1 ounce of raisin, and 4 ounces of cashews and sells for 50 cents. A box of mixture B contains 12 ounces of peanuts, 3 ounces of raisins, and 2 ounces of cashews and sells for 90 cents. He has available 5400 ounces of peanuts, 1200 ounces of raisins, and 2400 ounces of cashews. How many boxes of each mixture should he make to maximize revenue?