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Advanced Algebra and Trigonometry - Honors Linear Programming Worksheet 1. A 4-H Club member raises only geese (x) and pigs (y). She wants to raise no more than 16 animals total, including no more than 10 geese. She spends $15 to raise a goose and $45 to raise a pig, and has $540 available for this project. Each goose produces a profit of $6 and each pig a profit of $20. You are trying to compute a maximum profit. A) Find the equation of the objective function which is to be maximized. B) Determine the constraint inequalities (Hint: Don’t forget to include x > 0 and y > 0 since you can’t have neither a negative number of geese nor a negative number of pigs.) C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum profit, noting the number of geese and pigs it would take to create this maximum profit. 2. An office manager needs to purchase new filing cabinets. He knows that Phoenix (x) cabinets cost $40 each, require 6 square feet of floor space, and hold 8 cubic feet of files. On the other hand, each Firebird (y) cabinet costs $80, requires 8 square feet of floor space, and holds 12 cubic feet. His budget permits him to spend no more than $560 on files, while the office has room for no more than 72 square feet of cabinets. The manager desires the greatest storage capacity within the limitations imposed by funds and space. A) Find the equation of the objective function which is to be maximized. B) Determine the constraint inequalities (Hint: Including the trivial ones, there are four of them) C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the number of each which can be purchased to maximize storage space. 3. Certain farm animals must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients come from beef (x), which costs $0.18 per unit and supplies 2 grams of protein and 4 of fat, and cereal (y), with 6 grams of protein and 2 of fat, costing $0.12 per unit. Cereal is bought under a long-term contract requiring that at least 2 units of cereal be used per serving. The goal is to keep the cost a minimum. A) Find the equation of the objective function which is to be minimized. B) Determine the constraint inequalities (Hint: Including the trivial ones, there are five of them) C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the minimum cost per serving, noting the number of units of beef and cereal. 4. McDonald’s has been reserved for a special kids birthday party. A total of 150 people will attend this party. There will be two options for food: SUPER Happy Meal (x) and Happy Meal (y). The manager is told that there should be at least 30 more SUPER Happy Meals available than Happy Meals and that there should be at least 40 Happy Meals available. The manager has to pay $3.00 for its SUPER Happy Meals and $2.00 for its Happy Meals. A) Find the equation of the objective function which is to be minimized. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the minimum cost for the manager, noting the how many of each meal should be prepared. 5. Apple makes two models of iPOD’s…known in the industry as ‘basic’ (x) and ‘deluxe’ (y). The basic model requires 2 hours of assembly time and 24 square inches of material per unit; the deluxe model requires 2 hours of assembly time and 32 square inches of material per unit. Apple has available 600 hours of assembly time and 8,000 square inches of material per month. The profit on the ‘basic’ model is $100 per unit and $150 on the ‘deluxe’ unit. A) Find the equation of the objective function which is to be maximized. B) Determine the constraint inequalities C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum profit for Apple, noting the number of basic and deluxe units sold to maximize profit. 6. Target must ship at least 100 televisions to its two stores in Frankfort (x) and Orland Park (y). Each store can hold a maximum of 100 televisions. Frankfort has 25 televisions already, and Orland Park has 20. It costs $12 to ship a television to Frankfort and $10 to ship one to Orland Park. The goal is to minimize cost. A) Find the equation of the objective function which is to be minimized. B) Determine the constraint inequalities C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the minimum cost for Target, noting the number of televisions to be shipped to each store. 7. Fanny May needs to start thinking about Valentine’s Day candy. In each store, 60 pounds of chocolate and 100 pounds of mints are available to make 5-pound boxes of candy. A ‘regular box’ (x) has 4 pounds of chocolates and 1 pound of mints and sells for $10. A ‘deluxe box’ (y) has 2 pounds of chocolates and 3 pounds of mints and sells for $16. A) Find the equation of the objective function which will maximize revenue. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum revenue for Fanny May, noting the number of ‘regular boxes’ and ‘deluxe boxes’. 8. Microsoft is in the process of preparing Vista, it’s new operating system. The programming is done, and now it’s time to create the CD’s and box them. There are two versions of Vista: Home Edition (x) and Professional Edition (y) currently being produced. To get everything prepared for shipping requires a three machine process. Machine I (which makes the actual CD) takes 0.1 minutes for the Home Edition, and 0.1 minutes for the Professional edition. Production schedule allows for 240 minutes on this machine each day. Machine II (which gathers the manuals together) takes 0.1 minutes for the Home Edition, and 0.4 minutes for the Professional Edition. Production schedule allows for 720 minutes on this machine each day. Machine III (which places everything in the box itself) takes 0.1 minutes for the Home Edition, and 0.5 minutes for the Professional Edition. Production schedule allows for 800 minutes on this machine each day. The Home Edition sells for $100 and the Professional Edition sells for $120. A) Find the equation of the objective function which will maximize revenue. B) Determine the constraint inequalities C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum revenue for Microsoft, noting the number of ‘Home Editions’ and ‘Regular Editions.’ 9. A pension fund manager decides to invest at most $40 million in U.S. Treasury Bonds (x) paying 12% annual interest and in mutual funds (y) paying 8% annual interest. After reading an article in Money magazine, he now plans to follow their advice and invest at least $20 million in bonds and at least $15 in mutual funds. A) Find the equation of the objective function (in millions) which will maximize interest earned. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum amount of interest (in millions) he can earn for the pension fund, noting the amount invested in U.S. Treasure Bonds and the amount invested in mutual funds. 10. Mobil Oil receives oil products from two refineries…one in United States (x) and one in the Iran (y). Mobil needs a total of at least 6000 barrels of gasoline, 3000 barrels of kerosene, and 2000 barrels of fuel oil per week. The United States will supply 1000 barrels of gasoline, 1500 barrels of kerosene, and 500 barrels of fuel oil per day. Iran will supply 3000 barrels of gasoline, 500 barrels of kerosene and 500 barrels of fuel oil per day. The refinery in the United States charges $30,000 per day, and the refinery in Iran charges $40,000 per day. A) Find the equation of the objective function which will minimize costs. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the number of days per week Mobil should buy from each country in order to minimize costs. 11. A Mokena politician has budgeted $80,000 for her media campaign. She plans to distribute these funds between TV ads (x) and radio ads (y). Each 1-minute TV ad is expected to be seen by 20,000 viewers on the local cable channel, and each 1-minute WJOL radio ad is expected to be heard by 4000 listeners. Each minute of TV time costs $8000 and each minute of radio time costs $2000. She has been advised to use at most 90% of her media campaign budget on television ads. A) Find the equation of the objective function which will maximize the number of people she reaches. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the maximum number of people she can reach on her budget, noting the number of each type of ad. 12. Griffin Airlines wants to fly 1400 members of the ski club to Colorado. The airline owns two types of plane. The Boeing 707 (x) can carry only 50 passengers, requires 3 flight attendants, and costs $14,000 for the trip. The Boeing 767 (y) can carry 300 passengers, requires 4 flight attendants, and costs $90,000 for the trip. The airline must use lat least as many Boeing 707’s as Boeing 767’s and has available only 42 flight attendants. Griffin Airlines wants to minimize cost for the trip. A) Find the equation of the objective function which will maximize the number of each type of airplane to be used. B) Determine the constraint inequalities C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the number of each type of airplane necessary to minimize cost for the trip. 13. At the end of every month, after filling orders for its regular customers, a coffee company has some pure Colombian coffee and some special blend coffee remaining. The practice of the company has been to package a mixture of the two coffees into 16-ounce packages as follows: a low-grade mixture (x) containing 4 ounces of Colombian coffee and 12 ounces of special-blend coffee and a high-grade mixture (y) containing 8 ounces of Colombian and 8 ounces of special- blend coffee. A profit of $0.30 per package is made on the low-grade mixture, whereas a profit of $0.40 per package is made on the high-grade mixture. This month, 1920 pounds of special-blend coffee and 1600 pounds of pure Colombian coffee remain. The company is trying to maximize its profits. A) Find the equation of the objective function which will maximize the number of each grade coffee mixture to make. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the number of each mixture (low-grade and high grade) the company must make to maximize profits. 14. The Mokena Market combines ground beef (x) and ground pork (y) in a single package for meat loaf. The ground beef is 75% lean (75% beef, 25% fat) and costs the market $0.75 per pound. The ground pork is 60% lean and costs the market $0.45 per pound. The meatloaf must be at least 70% lean. The market wants to use at least 50 pounds of its available pork, but not more than 200 pounds of its available ground beef. The Mokena Meat Market is trying to minimize its costs. A) Find the equation of the objective function which will minimize the cost for the Mokena Meat Market. B) Determine the constraint inequalities. C) Sketch a graph of the system of inequalities, labeling the appropriate vertices of your feasible region. D) Find the number of pounds of ground beef and pork to be used which will minimize costs.