Math 3 Name______
Linear Programming Problems Date ______Block _____
For each problem you must:
1. Define the variables.
2. Write the objective function.
3. Write the constraints.
4. Graph the feasible region.
5. Find the coordinates of the vertices of the feasible region.
6. Evaluate the objective function at each vertex.
7. Answer the question in a complete sentence.
1) Baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. A tray of bran muffins requires 2 c milk and 3 c wheat flour. A baker has 16 c milk and 15 c wheat flour available. If each tray of corn muffins makes a profit of $3 and each tray of bran muffins makes a profit of $2, how many trays of each type of muffin should the baker make to maximize her profits?
2) For a school bake sale you are making whole-wheat bread and apple bran muffins. You will make a $35 profit on each batch of bread and a $10 profit on each batch of muffins. It will take you 4 hours to prepare and 1 hour to bake each batch of bread. It takes 0.5 hours to prepare and 0.5 hours to bake each batch of muffins. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should you make in order to maximize your profit?
3) The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 40 students, requires 3 chaperones, and costs $1,200 to rent. Each van can transport 8 students, requires 1 chaperone, and costs $100 to rent. Since there are 400 students in the senior class who are eligible to go on the trip, the officers must plan to accommodate at least 400 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What is the minimum transportation cost?
4) 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-hour from the cutting department and 0.8 labor-hour from the assembly department. Each four person boat requires 1.8 labor-hours from the cutting department and 1.2 labor-hours from the assembly department. The maximum labor-hours available per month in the cutting department is 864 and the maximum labor-hours available per month in the assembly department is 672. The company makes a profit of $25 on each two-person boat and $40 on each four-person boat. How many of each type of boat should be manufactured each month to maximize the profit?
5) Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city has $2100 to spend on planting spruce and maple trees and 45,000 square feet of land available for planting the trees. Each spruce tree costs $30 and requires 600 square feet of land. Each maple tree costs $40 and requires 900 square feet of land. If each spruce tree absorbs 650 lb of carbon dioxide per year and each maple tree absorbs 300 lbs of carbon dioxide per year, how many of each tree should the city plant to maximize carbon dioxide absorption?
6) A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of restaurants, drive-through and full-service. A drive-through restaurant costs $100,000 to construct, requires 5 employees, and has an expected annual revenue of $200,000. A full-service restaurant cost $150,000 to construct, requires 15 employees, and has and expected annual revenue of $500,000. The chain has $2,400,000 in capital available of expansion. Labor contracts require that they hire no more than 210 employees, and licensing restrictions require that they open no more than 20 new restaurants. How many restaurants of each type should the chain open in order to maximize the expected revenue? How much of their capital will they use and how many employees will they hire?
7) A laboratory technician in a medical research center is asked to formulate a diet from two commercially packaged foods, food A and food B, for a group of animals. Each ounce of food A contains 8 units of protein, 16 units of carbohydrate, and 2 units of fat. Each ounce of food B contains 4 units of protein, 32 units of carbohydrate, and 8 units of fat. The minimum daily requirements for carbohydrates and protein are 1,024 units of carbohydrate, and 176 units of protein. The maximum amount of fat allowed is 384 units. If food A costs $0.05 per ounce, and food B costs $0.07 per ounce, how many ounces of each food should be used to meet the minimum daily requirements at the least cost? What is the minimum cost?