Linear Programming Problems

  1. Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city has $2100 and 4500 ft2 of land to plant spruce and male trees. A spruce requires 600 ft2 and cost $30, while the maple requires 900 ft2 and cost $40. How many of each tree should the city plant to maximize carbon dioxide absorption? Hint: the city wants to plants as many trees as possible.
  1. Ish’s Butter Cookies sells large tins of butter cookies and small tins of butter cookies. The factory can prepare at most 200 tins of cookies a day. Each large tin of cookies requires 2 pounds of butter, and each small tin requires 1 pound of butter. There is a maximum of 300 pounds of butter available each day. The profit from each day’s cookie production is: $6 for the large tin and $4.80 for the small tin. Find the maximum profit that can be expected in a day.
  1. A car manufacturing company is asked to do a cost analysis to figure out which model car is cheaper to make. Next month they will order two different types of parts to two different models of cars, a more expensive model A and a less expensive model B. They must figure out how much of each model to order in order to minimize costs. They expect to sell at least 20 cars in the first month, some of model A and some of model B. Model A leaves $800 profit for the salesman and model B leaves a $400 profit for the salesman. Total profits must be at least $12000. The wholesale cost of model A is $32,000 and the wholesale cost of model B is $28,000.
  1. Central Delivery Service is near the beginning of its opening, but a problem has arisen. They are using small trucks and large trucks to transport their products in crates. All the crates are the same size. Everyday there is at most 10 truck drivers available. Each truck requires only one driver. The small trucks take 10 minutes to load and the large truck take 30 minutes each to load. The total loading time must not be more that 3 hours and only one truck can be loaded at a time. Each small truck carries 30 crates and each large truck carries 70 crates. How many of each truck should be used to maximize the number of crates to be transported each day.
  1. A designer of expensive leather jackets, created two new jacket designs for the new season: a long one and a short one. Each short leather jacket requires 1 labor hour from the cutting department and 3 labor hours from the sewing department. Each long leather jacket requires 2 labor hours from the cutting department and 4 labor hours from the sewing department. This designer is sharing cutting and sewing services with other designers, and as such, there are only 32 labor hours per week available in the cutting department and 84 labor hours per week available in the sewing department for him. In addition, because of the limited appeal of the long jacket, the distributor cannot take anymore than 12 long jackets per week. If the designer makes $50 profit on each short leather jacket and $80 on each long one, how many of each type of jacket should he have manufactured per week in order to maximize his profits?
  1. John has 150 acres of land suitable for cultivating crops A and B. The cost of cultivating crop A is $40/acre and the cost of crop B is $60/acre. The farmer has a maximum of $7200 available for land cultivation. Each acre of crop A requires 20 labor hours, and each acre of crop B requires 25 labor hours. The farmer has a maximum of 3300 labor hours available. If he expects to make a profit of $150/acre of crop A and $200/acre on crop B, how many acres of each crop should be plant in order to maximize his profit?
  1. At the Ant Green Use Car Lot, they sell Honda Accords and Honda Civics. The lot can not hold more that 50 cars. At an auction the paid $8000 for each Accord and $4000 for each Civic. They spent a total of $240,000. They make a profit of $800 on each Civic sold and $1000 on each Accord. How many of each car do they have to sell to maximize profit?
  1. Globocom Toys makes two video games, the Ending Dream and the Neo War using two computers, the IBM and the Linx. Each unit of the Ending Dream video that is produced requires 50 minutes processing time on the IBM and 30 minutes processing time on the Linx. Each unit of the Neo War video that is produced requires 25 minutes processing time on the IBM and 35 minutes processing time on the Linx. Available processing time on the IBM is 40 hours and 35 hours on the Linx. The price of the Ending Dream video is $75 and the price of the Neo War video is $95. Find the maximum profit of producing the videos.
  1. Mike’s Famous Toy Trucks manufactures two kinds of toy trucks-a standard model and a deluxe model. In the manufacturing process, each standard model requires 2 hours of grinding and 2 hours of finishing, and each deluxe model requires 2 hours of grinding and 4 hours of finishing. The company has 2 grinders and 3 finishers, each of whom work 40 hours per week. Each standard model toy truck brings a profit of $3 and each deluxe model truck bring a profit of $4. Assuming every truck made will be sold, how many of each should be made to maximize profit?
  1. A studio sells photographs and prints. It costs $20 to purchase each photograph and it takes 2 hours to frame it. It cost $25 to purchase each print and it takes 5 hours to frame it. The studio has at most $400 to spend and at most 60 hours to frame. It makes $30 profit on each photograph and $50 profit on each print. Find the number of each that the studio should purchase to maximize profit.
  1. A lunch counter sells 2 types of sandwiches, roast beef and chicken salad. The profit on the chicken salad sandwich is $2 and $3 for the roast beef. The amount of bread available is enough for 30 sandwiches. There are 4 hours available to prepare the sandwiches. If chicken salad sandwiches take 7 minutes to prepare and roast beef sandwiches take 10 minutes, how many of each type of sandwich should be prepared to maximize the profit?
  1. A manufacturer produces 2 models of mountain bikes. The times (in hours) required for assembling and painting model A are 5 and 2 respectively and for model B, 4 and 3. The maximum total weekly hours available in the assembly department is 200 hours and 180 hours for painting. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit?
  1. Stafford Manufacturing Inc. produces two models of calculators. They have a graphing calculator and a scientific calculator. Long-term demands for the models, mandates that the company manufacture at least 100 graphing calculators and at least 80 scientific calculators each day. Due to limitations on production capacity, no more that 200 graphing calculators and no more than 170 scientific calculators can be made daily. To satisfy the shipping contract, at total of at least 200 calculators must be shipped everyday. A scientific calculator costs $5 do produce, and graphing calculator costs $60 to produce. How many of each calculator should they produce each day to minimize cost?
  1. Mr. Davidoff owns a car and a motorcycle. He has at most 12 gallons of gasoline to be used between the car and the motorcycle. The car’s tank holds 10 gallons and the motorcycle’s tank holds 3 gallons. If the mileage for the car is 27 mpg and the motorcycle mileage is 98 mpg, how many gallons of gas should each vehicle use if Mr. Davidoff wants to travel as far as possible? What is the maximum number of miles?
  1. Four artists and three writers create two types of greeting cards. Each art card requires four hours of art and two hours of writing. Each sonnet card takes two hours of art and four hours of writing. Each employee can work up to 40 hours per week. The company makes a profit of $2 on each art card and $1 on each sonnet card. How many of each type of card should be created and sold to maximize profits?
  1. Cassandra is about to take a math test that contains short answer questions worth 4 points each and word problems worth 7 points each. She is required to do at least 5 short answer questions, but no more than 10. She must also do at least 3 word problems but no more than 10. If she is required to do no more than 18 problems in total, how many of each type should she do to maximize her score? What is the maximum score?
  1. The members of a girls sorority at UNCC are selling car decors in the school cafeteria to raise money for their house. A local print shop has donated what they need. The girls have enough license plates for 40 cars and enough bumper stickers for 60 cars. There are 90 people who want to buy. If they plan to sell each plate for $5 and each sticker for $3, and they sell to 90 people, what is the maximum profit they can expect to make?
  1. Andersons Tool Shed builds tool sheds. They use 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. They have available 60 sheets of dry wall and 135 studs. If we make $390 profit on a small shed and $520 on a large shed, how many of each type of building should they build to maximize profit?
  1. The Georgia Tech Tool Company uses 3 machines to manufacture 2 models of Buzz Mascots – the BobbleHead and the Stuffed Buzz. The BobbleHead requires 1 hour on machine A, 2 hours on machine B and 1.6 hours on machine C. The Stuffed Buzz requires 2 hours on machine A, 1 hour on machine B and 1.6 on machine C. Each machine can be used for at most 40 hours a week. If the profit on a BobbleHead is $7.45 and the profit on a Stuffed Buzz is $8.95, how many of each type should be made each week to maximize profit? What is the maximum profit?
  1. A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye cost $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes 1 hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye, how many acres of each should be planted to maximize profits?
  1. Fennig Farms has just bought 45 acres of land. Each acre can be planted with wheat or corn. Only 100 workers and 120 tons of fertilizer are available. An acre of wheat requires 3 workers and 2 tons of fertilizer. An acre of corn requires 2 workers and 4 tons of fertilizer. The revenue for each is $200 for wheat and $300 for corn. How can we maximize revenue?
  1. The area of a parking lot is 600 square meters. A car requires 6 square meters. A bus requires 30 square meters. The attendant can handle only 60 vehicles. If a car is charged $2.50 and a bus $7.50, how many of each should be accepted to maximize income?
  1. We are going to make 2 new styles of prom dresses for the new prom season: the sleeveless chiffon and the 2 piece taffeta. The chiffon dress requires 2 hours of sewing, and 2 hours of alterations. The taffeta dress requires 2 hours of sewing and 1 hour of alterations. There are no more than 24 hours available for sewing and no more than 20 hours available for alterations. The price of the chiffon dress is $375 and the price of the taffeta dress is $350. How many of each dress should be made to produce maximum income?
  1. Alltel makes two models of cell phones, the Razor and the Blade. Each Razor phone makes the company $10 in profit and each Blade makes $7. Both phones require human and machine time to produce. The Razor requires 4 hours of human labor and 2 hours of machine time. The Blade requires 3 hours of human labor and 3 hours of machine labor. Executives of the company use time management skills to allow 15 hours of human labor and 12 hours of machine labor. What combination of the two phones can be made to maximize profits?
  1. As office manager of her firm, Marcellyne has been directed to buy new filing cabinets. She knows that Cabinet A costs $10, requires 6 square feet of floor space and holds 9 cubic feet of files. Cabinet B costs $20 requires 8 square feet of floor space and holds 12.5 cubic feet of files. She can spend at most $140 and her office has room for no more than 72 square feet of cabinets. Her goal is to maximize storage capacity within her money and space limitations. How many of each type of cabinet should she buy, and how many cubic feet of storage will she get?
  1. My business is opening at two new locations. Both locations need marble on the floor of the main lobby. Each 1 x 1 piece of marble cost $5 and each 2 x 2 piece cost $8. There are 75 pieces of 1 x 1 marble and 60 piece of 2 x 2 marble. The combined two locations need 90 pieces or less to do both of the floors. What is the maximum amount I will need to spend?
  1. A small clothing company makes two styles of Georgia Tech shirts: a t-shirt and a tank top. To make a t-shirt, 10 minutes of cutting time and 30 minutes of sewing time and required. To make a tank top, 30 minutes of cutting time and 15 minutes of sewing time are required. Currently, at most 20 hours a day are available for cutting and at most 15 hours a day are available for sewing. Supposed they earn $13.50 profit for each t-shirt and $9.95 profit for each tank top they make. Assuming they can sell all the shirts they make, how many of each type should they produce in order to earn the most profit per day? What is the maximum daily profit?

28. Suppose you own an electronics store and are about to order a shipment of portable TVs and radios. Each TV weighs 10 lbs, and each radio weighs 3 lbs. The total shipment cannot weigh more than 75 lb. You must also order at least 5 radios and at least 3 TVs. If you make $10 for each radio you sell, and $20 for each TV sold, how many of each should you order and sell to maximize your profit?

29. A television manufacturer makes console and wide screen TVs. The profit per unit is $125 for the console TV and $200 for the wide-screen TVs. Equipment in the factory allows for making at most 450 console TVs and 200 wide-screen TVs in one month. The cost to manufacturer per unit is $600 for the console TVs and $900 for the wide-screen TVs. Total monthly costs cannot exceed $360,000. How many of each type of TV must you manufacture in order to maximize profits?

30. A pizza shop makes $1.50 on each small pizza and $2.15 on each large pizza. On a typical Friday, it sells between 70 and 90 small pizzas and 100 and 140 large pizzas. The total sales have never exceeded 210 pizzas. How many of each size pizza must be sold to maximize profits?