AMT Linear Programming Example
Roger is making hand-tooled leather belts and wallets to sell at a craft fair. He can bring no more than 20 items to the fair. Each belt takes 8 hours to make and sells for $45. Each wallet takes 2 hours to make and sells for $15. He has up to 70 hours to spend working on the leather items. He assumes that he will sell all the items that he makes. How many belts and wallets should he make in order to make the most money?
1. USE WORDS TO STATE WHAT EACH VARIABLE REPRESENTS:
x = ______
y = _______
2. WRITE A SYSTEM OF CONSTRAINTS (INEQUALITIES)
3. WRITE A LINEAR COMBINATION TO MAXIMIZE OR MINIMIZE
4. GRAPH CONSTRAINTS AND SHADE THE FEASIBLE REGION...USE GRAPH PAPER!
5. IDENTIFY THE VERTICES OF THE FEASIBLE REGION
6. SUBSTITUTE ALL VERTEX VALUES INTO THE LINEAR COMBINATION
7. IDENTIFY THE MAXIMUM OR MINIMUM VALUE AS APPROPRIATE
8. WRITE YOUR FINAL ANSWER IN SENTENCE FORM.
1. You are the assistant manager of an appliance store. Next month you will order two types of stereo systems. How many of each model should you order to minimize your cost?
· Model A: Your cost is $300 and your profit is $40
· Model B: Your cost is $400 and your profit is $60
· You expect a profit of at least $4800
· You expect to sell at least 100 units
2. A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150 and the profit on a chair is $65. How many tables and chairs should they make per day to maximize their profit? What is their maximum profit?
3. Your school has contracted with a professional magician to perform at the school. The school has guaranteed an attendance of at least 1000 and total ticket receipts of at least $4800. The tickets for students are $4 for students and $6 for non-students, of which the magician receives $2.50 and $4.50 profit respectively. What is the minimum amount of money the magician could receive?
4. A t-shirt company makes t-shirts and hoodies. They can make between 80 and 100 t-shirts in one day. They can produce between 50 and 80 hoodies in one day. They can make, at most, 160 total units in one day. If the profit on each t-shirt is $6 and the profit on each hoodie is $10, how many of each kind do they need to make a maximum profit? What will this maximum profit be?
5. A candy manufacturer has 130 pounds of chocolate-covered cherries and 170 pounds of chocolate-covered mints in stock. He decides to sell them in the form of two different mixtures. One mixture will contain half cherries and half mints by weight and will sell for $2.00 per pound. The other mixture will contain one-third cherries and two-thirds mints by weight and will sell for $1.25 per pound. How many pounds of each mixture should the candy manufacturer prepare in order to maximize his sales revenue?
6. The Osgood County refuse department runs two recycling centers. Center 1 costs $40 to run for an eight hour day. In a typical day 140 pounds of glass and 60 pounds of aluminum are deposited at Center 1. Center 2 costs $50 for an eight-hour day, with 100 pounds of glass and 180 pounds of aluminum deposited per day. The county has a commitment to deliver at least 1540 pounds of glass and 1440 pounds of aluminum per week to encourage a recycler to open up a plant in town. How many days per week should the county open each center to minimize its cost and still meet the recycler’s needs?