1. A pension fund manager decides to invest a total of at most $40million in U.S. Treasury bonds paying 5% annual interest and in mutual funds paying 7% annual interest. He plans to invest at least $5 million in bonds and at least $10 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $7000 on fees. How much should be invested in each to maximize annual interest? What is the maximum annual interest?
2. A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 2 units of carbohydrates, and 1 unit of fat, and will contain 20 calories. Each ounce of nuts will supply 3 units of protein, 1 unit of carbohydrate, and 2 units of fat, and will contain 30 calories. Every package must provide at least 9 units of protein, at least 8 units of carbohydrates, and no more than 10 units of fat. What is the least number of calories possible in a package.
3. A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 engines, and plant II needs at least 32 engines. The company can send at most 140 engines to these assembly plants. It costs $30 per engine to ship to plant I and $50 per engine to ship to plant II. Plant I gives the manufacturing company $20 in rebates toward its products for each engine they buy, while plant II gives similar $10 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?337
4. A population gathers plants and animals for survival. They need at least 260 units of energy, 300 units of protein, and 8 hides. One unit of plants provides 30 units of energy, 10 units of protein, and 0 hides. One animal provides 20 units of energy, 25 units of protein, and 1 hide. Only 13 units of plants and 16 animals are available. It costs 20 hours of labor to gather one unit of a plant and 10 hours for an animal. Find how many units of plants and animals should be gathered to meet the requirements with a minimum number of hours of labor.