5.4 The Irrational Numbers 257 47. 2 3 7 6 3 4 7 + − − 4 3 3 7 − − 48. 6 5 2 8 5 3 2 + − − 2 5 2 2 − − 49. 4 12 7 27 − 13 3 − 50. 2 20 3 45 − 5 5 − 51. 5 3 7 12 3 75 + − 4 3 52. 3 20 45 2 125 − + 13 5 In Exercises 53– 60, perform the indicated operation. Simplify the answer when possible. 53. 3 27 9 54. 5 10 5 2 55. 3 21 3 7 56. 10 35 5 14 57. 20 5 2 58. 125 5 5 59. 72 8 3 60. 145 5 29 In Exercises 61– 68, rationalize the denominator. 61. 1 7 7 7 62. 5 5 5 63. 2 3 6 3 64. 5 2 10 2 65. 20 3 2 15 3 66. 50 14 5 7 7 67. 10 6 15 3 68. 2 27 6 9 Problem Solving Approximating Radicals The following diagram is a sketch of a 16 in. - ruler marked using 1 2 inches. 1 2 3 4 5 6 7 8 9 101112131415 In Exercises 69–74, without using a calculator, indicate between which two adjacent ruler marks each of the following irrational numbers will fall. Support your answer by obtaining an approximation with a calculator. 69. 37 in. * 70. 61 in. * 71. 97 in. * 72. 123 in. * 73. 170 in. * 74. 200 in. * 75. Dropping an Object The formula t d 4 = can be used to estimate the time, t, in seconds it takes for an object dropped to travel d feet. Estimate the time, to the nearest tenth of a second, it takes for an object to drop a) 100 ft. 2.5 sec b) 400 ft. 5 sec c) 900 ft. 7.5 sec d) 1600 ft. 10 sec 76. Estimating Speed of a Vehicle The speed that a vehicle was traveling, s, in miles per hour, when the brakes were first applied, can be estimated using the formula s d 0.04 = where d is the length of the vehicle’s skid marks, in feet. Estimate the speed of a car that makes skid marks a) 4 ft long. 10 mph b) 16 ft long. 20 mph c) 64 ft long. 40 mph d) 256 ft long. 80 mph 77. Estimating Height The median height of male children less than 5 years old can be estimated using the formula H x 2.9 20.1, = + where H is the height in inches and x is the child’s age in months. Estimate the median height, to the nearest tenth of an inch, of a male child whose age is a) 24 months 34.3 inches b) 30 months. 36.0 inches c) 36 months. 37.5 inches d) 42 months. 38.9 inches 78. A Swinging Pendulum The time T required for a pendulum to swing back and forth may be determined by the formula T l 2 980 π = where l is the length of the pendulum in centimeters (cm). Estimate the time, to the nearest tenth of a second, for the pendulum to swing back and forth if the length of the pendulum is a) 30 cm. 1.1 sec b) 35 cm. 1.2 sec c) 45 cm. 1.3 sec d) 55 cm. 1.5 sec Concept/Writing Exercises In Exercises 79–84, determine whether the statement is true or false. Rewrite each false statement to make it a true statement. A false statement can be modified in more than one way to be made a true statement. 79. c is a rational number for any composite number c. * 80. p is a rational number for any prime number p. False. p is an irrational number for any prime number p. 81. The sum of any two rational numbers is always a rational number. True *See Instructor Answer Appendix SergiyN/Shutterstock
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