288 CHAPTER 5 Number Theory and the Real Number System 25. a) To what decimal value is + ( 5 1)/2 approximately equal? 1.618 b) To what decimal value is − ( 5 1)/2 approximately equal? 0.618 c) By how much do the results in parts (a) and (b) differ? 1 26. Determine the ratio of the length to the width of your cell phone screen and compare this ratio to .Φ Answers will vary. 27. Determine the ratio of the length to width of various photographs and compare these ratios to .Φ Answers will vary. 28. Find three physical objects whose dimensions are very close to a golden rectangle. a) List the objects and record the dimensions. b) Compute the ratios of their lengths to their widths. c) Determine the difference between the golden ratio and the ratio you obtain in part (b)—to the nearest tenth— for each object. Answers will vary. Concept/Writing Exercises 29. The eleventh Fibonacci number is 89. Examine the first six digits in the decimal expression of its reciprocal, . 1 89 What do you determine? The decimal expansion shows several terms of the Fibonacci sequence. 30. Determine the ratio of the second to the first term of the Fibonacci sequence. Then determine the ratio of the third to the second term of the sequence and determine whether this ratio was an increase or decrease from the first ratio. Continue this process for 10 ratios and then make a conjecture regarding the increasing or decreasing values in consecutive ratios. * 31. A musical composition is described as follows. Explain why this piece is based on the golden ratio. Entire Composition 34 measures 55 measures 21 measures 34 measures Theme Fast, Loud Slow Repeat of theme The ratio of the second to the first, and the fourth to the third measures estimates the golden ratio. Challenge Problems/Group Activities 32. Draw a line of length 5 in. Determine and mark the point on the line that will create the golden ratio. Explain how you determined your answer. Answers will vary. 33. The divine proportion is + = a b a a b ( )/ / (see Fig. 5.13), which can be written + = b a a b 1 ( / ) / . Now let = x a b/ , which gives + = x x 1 (1/ ) . Multiply both sides of this equation by x to get a quadratic equation and then use the quadratic formula (Section 6.9) to show that one answer is = + x (1 5)/2 (the golden ratio). Answers will vary. 34. Pythagorean Triples A Pythagorean triple is a set of three whole numbers, a b c , , , { } such that a b c . 2 2 2 + = For example, since 6 8 (10) , 6, 8, 10 2 2 2 { } + = is a Pythagorean triple. The following steps show how to determine Pythagorean triples using any four consecutive Fibonacci numbers. Here we will demonstrate the process with the Fibonacci numbers 3, 5, 8, and 13. 1) Determine the product of 2 and the two inner Fibonacci numbers. We have 2(5)(8) 80, = which is the first number in the Pythagorean triple. So a 80. = 2) Determine the product of the two outer numbers. We have 3(13) 39, = which is the second number in the Pythagorean triple. So b 39. = 3) Determine the sum of the squares of the inner two numbers. We have 5 8 25 64 89, 2 2 + = + = which is the third number in the Pythagorean triple. So c 89. = This process has produced the Pythagorean triple, 80, 39, 89 . { } To verify, ( ) + = + = = (80) (39) 89 6400 1521 7921 7921 7921 2 2 2 Use this process to produce four other Pythagorean triples. Answers will vary. 35. Reflections When two panes of glass are placed face to face, four interior reflective surfaces exist, labeled 1, 2, 3, and 4. If light is not reflected, it has just one path through the glass (see the following figure). If it has one reflection, it can be reflected in two ways. If it has two reflections, it can be reflected in three ways. Use this information to answer parts (a) through (c). 0 reflections 1 path 1 reflection 2 paths 2 reflections 3 paths 1 2 3 4 a) If a ray is reflected three times, there are five paths it can follow. Show the paths. Answers will vary. b) If a ray is reflected four times, there are eight paths it can follow. Show the paths. Answers will vary. c) How many paths can a ray follow if it is reflected five times? Explain how you determined your answer. Answers will vary. Research Activities 36. Fibonacci Write a report on the history and mathematical contributions of Fibonacci. 37. Digits The digits 1 through 9 have evolved considerably since they appeared in Fibonacci’s book Liber Abacci. Write a report tracing the history of the evolution of the digits 1 through 9 since Fibonacci’s time. 38. Golden Ratio and Golden Rectangle Write a report indicating where the golden ratio and golden rectangle have been used in art and architecture. You may wish to include information on art and architecture related to the golden ratio and Fibonacci sequences. *See Instructor Answer Appendix
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