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Inequalities

Problems

Linear Programming

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A factory produces Brand A dog food and Brand B dog food. It must produce no more than 4 tons of Brand A every day, and at least 2 tons of Brand B every day. The factory must produce at least 4 tons of Brand A for every 5 tons of Brand B (the ratio of Brand A to Brand B must be at least , and the factory cannot produce more than 8 tons of total dog food in a day.

Brand A costs 1 thousand dollars per ton to make, and sells for 2 thousand dollars per ton. Brand B costs 2.5 thousand dollars to make, and sells for 3 thousand dollars per ton. Thus, the equation for revenue is R = 2x + 3y , the equation for cost is C = x + 2.5y , and the equation for profit is P = R - C = x + 0.5y .

Problem : How much of each brand should the factory make to maximize revenue?

tons of Brand A, and tons of Brand B.

Problem : How much of each brand should the factory make to minimize cost?

tons of Brand A, and 2 tons of Brand B.

Problem : How much of each brand should the factory make to maximize profit?

4 tons of Brand A, and 4 tons of Brand B.

The school cafeteria carries hot dogs and hamburgers. They cannot make more than 400 hot dogs in a single day. The cafeteria must make at least half as many hamburgers as hot dogs, but cannot make more than one-fourth the number of hot dogs plus 200. In addition, the number of hot dogs plus twice the number of hamburgers must be at least 400.

Hot dogs cost 10 cents to make, and the cafeteria sells them for 20 cents each. Hamburgers cost 60 cents to make, and the cafeteria sells them for 50 cents each.

Problem : How many of each item should the cafeteria make in order to have the highest revenue? What is the maximum amount of revenue the cafeteria can make?

400 hot dogs, 300 hamburgers. Revenue = $230.

Problem : How many of each item should the cafeteria make in order to minimize the cost? What is the minimum amount of money the cafeteria must spend?

200 hot dogs, 100 hamburgers. Cost = $80.

Problem : How many of each item should the cafeteria make in order to maximize the profit? What is the maximum profit?

400 hot dogs, 200 hamburgers. Profit = $20.

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