840 CHAPTER 11 Systems of Equations and Inequalities Financial Planning Raj recently graduated from college and received a signing bonus of $2500 from his employer, which he will invest. As the financial adviser, you recommend that Raj place at least $1500 in Treasury bills yielding 2% and at most $500 in corporate bonds yielding 3%. Develop a model that can be used to determine how much money Raj should place in each investment so that income is maximized. EXAMPLE 1 11.8 Linear Programming OBJECTIVES 1 Set Up a Linear Programming Problem (p. 840) 2 Solve a Linear Programming Problem (p. 841) Historically, linear programming evolved as a technique for solving problems involving resource allocation of goods and materials for the U.S. Air Force during World War II.Today, linear programming techniques are used to solve a wide variety of problems, such as optimizing airline scheduling. Most practical linear programming problems involve systems of several hundred linear inequalities containing several hundred variables. We will limit our discussion to problems containing only two variables, because we can solve such problems using graphing techniques.* 1 Set Up a Linear Programming Problem Let’s begin by returning to Example 12 from Section 11.7. * The simplex method is a way to solve linear programming problems involving many inequalities and variables. Developed by George Dantzig in 1946, it is particularly well suited for computerization. In 1984, Narendra Karmarkar of Bell Laboratories discovered a way to improve the simplex method. 65. Use the Intermediate Value Theorem to show that f x x x 6 5 6 2 ( ) = + − has a real zero on the interval 1, 2 . [ ] − 66. Solve the equation 2 cos cos 1 0 2 θ θ − − = for 0 2 . θ π ≤ < 67. Solve: x x x 2 4 3 18 − ≤− + ≤ + 68. If $7500 is invested in an account paying 3.25% interest compounded daily, how much money will be in the account after 5 years? 69. The horsepower P needed to propel a boat through water is directly proportional to the cube of the boat’ s speed s. If a boat needs 150 horsepower to travel 12 miles per hour, what horsepower does it need to travel 6 miles per hour? 70. Change y x log5 = to an equivalent statement involving an exponent. 71. Given f x x 2 5 ( ) = − and g x x 2, ( ) = + find the domain of f g x . ( )( ) 72. Consider the functions f x x x x 7 5 4 3 2 ( ) = − − + and f x x x 3 14 5. 2 ( ) ′ = − − Given that f is increasing where f x 0 ( ) ′ > and f is decreasing where f x 0, ( ) ′ < find where f is increasing and where f is decreasing. Because polynomials are continuous over its domain, all endpoints are included in the interval describing increasing/decreasing. In general, however, the numbers at the endpoints must be tested separately to determine if they should be included in the interval describing where a function is increasing or decreasing. ‘Are You Prepared?’ Answers 1. { } < x x 1 or ( ) −∞, 1 2. x y 22 22 2 2 (2, 0) (0,23) 3. x y 25 5 5 25 (23, 0) (0, 3) (0, 23) (3, 0) 4. x y 25 5 8 22 (21, 5) (0, 4) (1, 5) 5. True 6. { } [ ] − ≤ ≤ − x x 3 3 or 3, 3
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