Examples
1. A small firm manufactures necklaces and bracelets. The combined number of necklaces and bracelets that it can handle per day is 24. The bracelet takes 1 hour of labor to make and the necklace takes a half hour. The total number of hours of labor available per day is 16. If the profit on the bracelet is $2 and the profit on the necklace is $1, how many of each product should be produced daily to maximize profit?
2. Two chemical plants produce three grades (A, B, and C) of a certain chemical in a single operation, so the various grades are in fixed proportions. Assume Plant 1 produces 1 unit of A, 2 units of B, and 3 units of C in a single operation, and it charges $300 for what is produced in one operation. Assume one operation at Plant 2 costs the consumer $500 and produces 1 unit of A, 5 units of B, and 1 unit of C. A consumer needs 100 units of A, 260 units of B, and 180 units of C. How should the consumer place to orders so that costs are minimized?
Practice Exercises
1. A manufactures two types of electric hedge-trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packing department can package only 300 trimmers per day. If the company sells the cord-type model for $30 and the cordless model for $ 40, how many of each type should it produce per day to maximize its sales?
2. In problem 1, if the profit on each type is $10, how many of each type should the company produce per day to maximize its profit? Can its profit be maximized at more than one point in this case?
3. Two factories produce three different types of kitchen appliances. The table below summarizes the production capacity, the number of each type of appliance ordered, and the daily operating costs for the factories. How many days should each factory operate to fill the orders at minimum cost?
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Factory 1 Factory 2 Number Ordered
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Appliance 1 80 per day 20 per day 1600
Appliance 2 10 per day 10 per day 500
Appliance 3 20 per day 70 per day 2000
Daily cost $10,000 $20,000
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4. A candidate wishes to use a combination of radio and television advertisements in his campaign. Research has shown that each 1-minute spot on television reaches 0.09 million people and each 1-minute spot on radio reaches 0.006 million. The candidate feels he must reach 2.1 million people, and he must buy a total of 80 minutes of advertisements. How many minutes of each medium should be used to minimize costs if television costs $500/minute and radio costs $100/minute?
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