Linear Programming The Graphical Method Review Problems Finite

Linear Programming The Graphical Method Review Problems Finite 3 -R

• This whole test is completed without the aid of a chromebook • You may use a calculator on the whole test • This is an 80 point test Test Information

1 Review Problems

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11 Review Problems

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 3 x – 2 y for this region. x ≤ 5 y ≤ 4 x + y ≥ 2 The maximum value is 21 at (5, – 3). The minimum value is – 14 at (– 2, 4). 12 Review Problems

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 2 x + 3 y for this region. –x + 2 y ≤ 2 x – 2 y ≤ 4 x + y ≥ – 2 The vertices are at (– 2, 0) and (0, – 2). There is no maximum value. The minimum value is – 6 at (0, – 2). 13 Review Problems

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• The Cruiser Bicycle Company makes two styles of bicycles: the Traveler, which sells for $300, and the Tourister, which sells $600. Each bicycle has the same frame and tires, but the assembly and painting time required for the Traveler is only 1 hour, while it is 3 hours for the Tourister. There are 300 frames and 360 hours of labor available for production. How many bicycles of each model should be produced to maximize revenue? 23 Review Problems

• TV Electronics Inc. makes console and wide-screen televisions. The equipment in the factory allows for manufacturing at most 450 console televisions and 200 widescreen televisions in one month. The chart below shows the cost of making each type of television, as well as the profit for each. During the month of November, the company can spend $360, 000 to make these televisions. How many of each type should be produced in order to maximize profit? What is the maximum profit? 300 Consoles and 200 Widescreens for $77, 500 24 Review Problems

• Stitches Inc. can make at most 30 jean jackets and 20 leather jackets in a week. It takes a worker 10 hours to make a jean jacket and 20 hours to make a leather jacket. The total number of hours by all of the employees can be no more than 500 hours per week. The profit on a jean jacket is $20, and the profit on a leather jacket is $50. How many of each type should be produced in order to maximize profit? What is the maximum profit? 10 Jean Jackets and 20 Leather Jackets for $1200 25 Review Problems

• You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume? you should obtain a maximal volume of 100 cubic feet by buying eight of model X and three of model Y 26 Review Problems

• In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protien. But the rabbits should be fed no more than five ounces of food a day. Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0. 20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0. 30 per ounce. What is the optimal blend? you should get a minimum cost of sixty cents per daily serving , using three ounces of Food X only. 27 Review Problems

• A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. • If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? 100 scientific calculators and 170 graphing calculators 28 Review Problems

• A store sells two types of toys, A and B. The store owner pays $8 and $14 for each one unit of toy A and B respectively. One unit of toys A yields a profit of $2 while a unit of toys B yields a profit of $3. The store owner estimates that no more than 2000 toys will be sold every month and he does not plan to invest more than $20, 000 in inventory of these toys. How many units of each type of toys should be stocked in order to maximize his monthly total profit? The store owner has to have 1333 toys of type A and 667 toys of type B in order to maximize his profit. 29 Review Problems

1. 5. 2. 6. 3. 7. 4. 8. ANSWERS

13. 9. The vertices are at (– 2, 0) and (0, – 2). There is no maximum value. The minimum value is – 6 at (0, – 2). 14. 10. 15. 16. 11. 17. 18. 12. The maximum value is 21 at (5, – 3). The minimum value is – 14 at (– 2, 4). ANSWERS

19. 26. You should obtain a maximal volume of 100 cubic feet by buying eight of model X and three of model Y 20. 27. You should get a minimum cost of sixty cents per daily serving , using three ounces of Food X only. 21. 22. 28. 100 scientific calculators and 170 graphing calculators 23. 29. The store owner has to have 24. 300 Consoles and 200 Widescreens for $77, 500 1333 toys of type A and 667 toys of type B in order to maximize his profit. 25. 10 Jean Jackets and 20 Leather Jackets for $1200 ANSWERS