LINEAR PROGRAMMING PROJECT PACKET

Transcription

1 AQR-Q1 LINEAR PROGRAMMING PROJECT PACKET Name: BELLRINGERà Evaluate P = 20x + 30y for the coordinates 3, 5, 4, 7, and (5, 8). Which coordinate yields the greatest value of P? WHAT IS LINEAR PROGRAMMING? à Linear Programming: A method used to find such as a or a value that satisfies all of the of a. Constraints: The contained in the problem Feasible Region: The to the set of Objective Function: The to be or Corner-Point Principle: The and values of the each occur at one of the of the. GENERAL METHOD for solving linear programming problemsà 1. Write a system of linear inequalities to represent the (don t forget the real-life constraints!) 2. Graph the (using x and y intercepts) and shade the 3. Name each of the feasible region with a capital letter 4. Find the coordinates of the corner points/vertices of the feasible region 5. Write the to be maximized or minimized 6. Make a table with the corner points and evaluate the objective function at the coordinates of the corner points of the feasible region 7. Identify the corner point that produces the required maximum or minimum 8. State your results in a (a complete sentence) 9. (only do this step when asked) Substitute your maximum or minimum value from your table in your objective function, solve for y, and graph it.

2 EXAMPLEà A lawn and garden store sells gas-powered and electric hedge clippers. The store wants to sell no more than total hedge clippers per month and must be electric. The profit on each electric model is $ and the profit on each gas-powered model is $. The shop wants to earn the maximum profit per month on the sale of hedge clippers. Let x = the number of electric hedge clippers Let y = the number of gas powered hedge clippers

3 PRACTICEà A calculator company produces scientific calculators and graphing calculators. At least 100 but no more than 200 scientific calculators can be made daily, and at least 80 but no more than 170 graphing calculators can me made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. If each scientific calculator produces a $3 profit and each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize profits? 1) CONSTRAINTS: Assign a variable to represent the scientific calculators: Assign a variable to represent the graphing calculators: Can you produce negative calculators? What inequalities are suggested by your answer to the above question? and t t t 2) GRAPH FEASIBLE REGION: 3) FIND VERTICES/CORNER POINTS: 4) OBJECTIVE FUNCTION for Revenue: 5) EVALUATE OBJECTIVE FUNCTION AT CORNER POINTS: Vertex: Objective Function: 6) STATE ANSWER/GRAPH OBJECTIVE FUNCTION: Upon testing your corner points, where does the maximum Revenue occur (ordered pair)? (, ) How many scientific calculators and how many graphing calculators should be sold?

4 GROUP WORKà PROBLEM A: A ticket office sells reserved tickets and general-admission tickets to a concert. The auditorium normally holds no more than 5000 people. There can be no less than 1000 and no more than 3000 reserved tickets sold and each makes a profit of $12. There can be no more than 4000 general-admission tickets sold and each makes a $7 profit. The ticket office wants to maximize the profit on ticket sales. Find the maximum profit for the given constraints.

5 PROBLEM B: A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater. Find the maximum profit for the given constraints.

6 PROBLEM C: A farmer has 90 acres available for planting wheat and cotton. Seed costs $4 per acre for wheat and $6 per acre for cotton. Labor costs are $20 per acre for wheat and $10 per acre for cotton. The farmer intends to spend no more than $480 for seed and $1400 for labor. The expected income is $110 per acre for wheat and $150 per acre for cotton. Find the maximum income for the given constraints.

7 PROBLEM D: Some students want to start a business that cleans and polishes cars. It takes 1.5 hours of labor to clean a car and 2 hours of labor to polish a car. It costs $2.25 in supplies to clean a car and $4.50 in supplies to polish a car. The students can work a total of 120 hours in one week. They also decide that they want to spend no more than $160 per week on supplies. The students expect to make a profit of $7.75 for each car that they clean and a profit of $8.50 for each car that they polish. Find the maximum profit that the students can make.

8 PROBLEM E: The Cruiser Bicycle Company makes two styles of bicycles: the Traveler, which sells for $300, and the Tourister, which sells for $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?

9 INDIVIDUAL WORKà Please use your time with the computer and with Desmos graphing to perfect anything from above. With that complete, please move on to the problems below. 1. Given the following objective function and constraints graph the constraints and determine the maximum and minimum the maximum values. and minimum values. C 4x2y Constraints: x 0 y 0 2x y 8 x y 2 Constraints: x 0 y 0 2x y 8 x y 2 What are the vertices of the feasible region? Maximum Value Maximum Point Minimum Value Minimum Point 2. Given the objective function and the graphed feasible region, determine the maximum and minimum values. Objective function: T 2x y What are the vertices of the feasible region? Maximum Value: Minimum Value: 3. The vertices of a feasible region for a linear programming problem are : 1, 5, 2, 6,6, 3. objective function is T 2x3y. 8,1, Determine the maximum and minimum values and points given that the Maximum Value Maximum Point Minimum Value Minimum Point

10 All work is to be done on separate paper. The work needs to include the constraints, the function to maximize or minimize, the graph, and the vertices. Make sure you answer the question appropriately as well. Please define or identify your variables and label each axis on your graph. 4. 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? 5. The B & W Leather Company wants to add handmade belts and wallets to its product line. Each belt nets the company $18 in profit, and each wallet nets $12. Both belts and wallets require cutting and sewing. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours of cutting time and 3 hours of sewing time. If the cutting machine is available 12 hours a week and the sewing machine is available 18 hours per week, what ratio of belts and wallets will produce the most profit within the constraints? 6. A diet is to include at least 140 milligrams of Vitamin A and at least 145 milligrams of Vitamin B. These requirements can be obtained from two types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. If type X food costs $12 per pound and type Y food costs $8 per pound how many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost?