SECTION 4.7 Polynomial and Rational Inequalities 265 mass of the moon is × 7.349 1022 kilograms, and the mean distance from Earth to the moon is 384,400 kilometers. For an object between Earth and the moon, how far from Earth is the force on the object due to the moon greater than the force on the object due to Earth? Source: www.solarviews.com; en.wikipedia.org 75. Challenge Problem Bungee Jumping Originating on Pentecost Island in the Pacific, the practice of a person jumping from a high place harnessed to a flexible attachment was introduced to Western culture in 1979 by the Oxford University Dangerous Sport Club. One important parameter to know before attempting a bungee jump is the amount the cord will stretch at the bottom of the fall. The stiffness of the cord is related to the amount of stretch by the equation ( ) = + K W S L S 2 2 where = W weight of the jumper (pounds) = K cord’s stiffness (pounds per foot) = L free length of the cord (feet) = S stretch (feet) (a) A 150-pound person plans to jump off a ledge attached to a cord of length 42 feet. If the stiffness of the cord is no less than 16 pounds per foot, how much will the cord stretch? (b) If safety requirements will not permit the jumper to get any closer than 3 feet to the ground, what is the minimum height required for the ledge in part (a)? Source: American Institute of Physics, Physics News Update, No. 150, November 5, 1993. 71. Average Cost Suppose that the daily cost C of manufacturing bicycles is given by ( ) = + C x x 80 5000. Then the average daily cost C is given by ( ) = + C x x x 80 5000 . How many bicycles must be produced each day for the average cost to be no more than $100? 72. Average Cost See Problem 71. Suppose that the government imposes a $1000-per-day tax on the bicycle manufacturer so that the daily cost C of manufacturing x bicycles is now given by ( ) = + C x x 80 6000. Now the average daily cost C is given by ( ) = + C x x x 80 6000 . How many bicycles must be produced each day for the average cost to be no more than $100? 73. Field Trip Mrs.West has decided to take her fifth grade class to a play. The manager of the theater agreed to discount the regular $40 price of the ticket by $0.20 for each ticket sold. The cost of the bus, $500, will be split equally among the students. How many students must attend to keep the cost per student at or below $40? 74. Challenge Problem Gravitational Force According to Newton’s Law of Universal Gravitation, the attractive force F between two bodies is given by = F G mm r 1 2 2 where = m m , themasses 1 2 of the two bodies = r distance between the two bodies = G gravitational constant = × − 6.6742 10 11 ⋅ ⋅ − newtonsmeter kilogram 2 2 Suppose an object is traveling directly from Earth to the moon. The mass of Earth is × 5.9742 1024 kilograms, the 76. The inequality + <− x 1 5 4 has no solution. Explain why. 77. A student attempted to solve the inequality + − ≤ x x 4 3 0 by multiplying both sides of the inequality by −x 3 to get + ≤ x 4 0. This led to a solution of { } ≤− x x 4 . Is the student correct? Explain. Explaining Concepts 78. Write a rational inequality whose solution set is { } − < ≤ x x 3 5 . 79. Make up an inequality that has no solution. Make up one that has exactly one solution. 86. Determine whether the graph of ( ) ( ) + − = + x y x x y 2 9 2 2 2 2 2 is symmetric with respect to the x-axis, y-axis, origin, or none of these. 87. Approximate the turning points of ( ) = − + f x x x2 4. 3 2 Round answers to two decimal places. 88. Solve: − = + + x x x 5 3 2 11 1 2 2 89. What are the quotient and remainder when − + x x 8 4 5 2 is divided by +x4 1? 80. Solve: − ≤ + x x 9 2 4 1 81. Factor completely: + − x y x y x y 6 3 18 4 4 3 5 2 6 82. Suppose y varies directly with x. Write a general formula to describe the variation if = y 2 when = x 9. 83. If ( ) = − f x x3 1 and ( ) = + g x x3 1, find ( )( ) ⋅ f g x and state its domain. 84. If ( ) = + f x x4 3, find ( ) − f x 3 4 . 85. Solve ω = LC 1 for C. Retain Your Knowledge Problems 80–89 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 1. { } ( ) <− −∞ − x x 1 2 or , 1 2 21 2 2 2 0 1 1 – 2 2. { } [ ] − ≤ ≤ − x x 3 8 or 3,8 –3 0 8 ‘Are You Prepared?’ Answers
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