6.1 Golden Section 6.2 More about Exponential and Logarithmic Functions 6.3 Nine-Point Circle Contents 6 Further Applications (1)

Slides:



Advertisements
Similar presentations
Biography ( ) Fibonacci is a short for the Latin "filius Bonacci" which means "the son of Bonacci" but his full name was Leonardo of Pisa, or Leonardo.
Advertisements

Exponential Growth and Decay
Phi, Fibonacci, and 666 Jay Dolan. Derivation of Phi “A is to B as B is to C, where A is 161.8% of B and B is 161.8% of C, and B is 61.8% of A and C is.
Original Question: How fast rabbits can rabbits breed in ideal circumstances? Suppose a newly-born pair of.
THE FIBONOCCI SEQUENCE IN REAL LIFE BY ANNE-MARIE PIETERSMA, HARRY BUI, QUINN CASHELL, AND KWANGGEUN HAN.
Lecture 3, Tuesday, Aug. 29. Chapter 2: Single species growth models, continued 2.1. Linear difference equations, Fibonacci number and golden ratio. Required.
MATH 109 Exam 2 Review. Jeopardy Show Me The $$$$$ Potent Potables Famous Log Cabins Captain’s Log Potpourri
Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds.
EXCURSIONS IN MODERN MATHEMATICS SIXTH EDITION Peter Tannenbaum 1.
Whiteboardmaths.com © 2004 All rights reserved
Fibonacci Numbers.
FIBONACCI NUMBERS 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, ,
Are We Golden? Investigating Mathematics in Nature
Great Theoretical Ideas in Computer Science.
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.8 Fibonacci Sequence.
5. 6. Further Applications and Modeling with
Applied Discrete Mathematics Week 9: Relations
Aim: How do we solve verbal problems leading to exponential or logarithmic equations? Do Now: Jacob has 18 kg of radium. If the half-life of radium is.
Sequences defined recursively
© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models.
Exponential Growth & Decay Modeling Data Objectives –Model exponential growth & decay –Model data with exponential & logarithmic functions. 1.
Logarithmic, Exponential, and Other Transcendental Functions Copyright © Cengage Learning. All rights reserved.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2013, 2009, 2005 Pearson Education,
SECTION 5-5 The Fibonacci Sequence and the Golden Ratio Slide
Are You Perfect? Writing Prompt: What do you consider perfect?
GOLDEN MEAN AUKSO PJŪVIS. Definition of the Golden Rectangle The Golden Rectangle is a rectangle that can be split into a square and a rectangle similar.
The Golden Ratio and Fibonacci Numbers in Nature
Date: 3 rd Mar, 2011 Time: 11:59:59 Venue: Class: Math 162 Follow Me 1.
8-1 Exploring Exponential Models Exponential Growth Growth Factor Growth Factor b > 1 Exponential Decay Decay Factor Decay Factor 0 < b < 1 Growth Rate.
Lesson 9-4 Exponential Growth and Decay. Generally these take on the form Where p 0 is the initial condition at time t= 0 population shrinking  decay.
5 CM 4 CM Calculation Area = Length times Width (lw or l x W) Note Length of a rectangle is always the longest side.
CSE 2813 Discrete Structures Recurrence Relations Section 6.1.
Applications and Models: Growth and Decay
Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and series? Find the next three numbers in the sequence 1,
UNIT 5: EXPONENTIAL GROWTH AND DECAY CONTINUOUS Exponential Growth and Decay Percent of change is continuously occurring during the period of time (yearly,
Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Scientists used carbon-14 dating of the soot from the oil lamps the artists used for lighting.
Fibonacci Sequences and the Golden Ratio Carl Wozniak Northern Michigan University.
GOLDEN RATIO GOLDEN SECTION FIBONACCI NUMBERS 1, 1, 2, 3, 5, 8, 13….. The ratio of any consecutive numbers is the golden ratio A pattern found in nature.
MTH 112 Section 3.5 Exponential Growth & Decay Modeling Data.
Section 6 Chapter Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives Exponential and Logarithmic Equations; Further Applications.
CONTINUOUS Exponential Growth and Decay
Do Now How long would it take for an initial deposit of $1000 to grow into $1500 if you deposit it into an account that earns 4% interest compounded monthly?
Agenda Lecture Content:  Recurrence Relations  Solving Recurrence Relations  Iteration  Linear homogenous recurrence relation of order k with constant.
Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci.
Background Knowledge Write the equation of the line with a slope of ½ that goes through the point (8, 17)
Lesson 3.5, page 422 Exponential Growth & Decay Objective: To apply models of exponential growth and decay.
The Fibonacci Sequence
By Steven Cornell.  Was created by Leonardo Pisano Bogollo.  It show’s the growth of an idealized rabbit population.
Exponential Growth and Decay. Exponential Growth When you have exponential growth, the numbers are getting large very quickly. The “b” in your exponential.
DO NOW HW: Exponential Functions Worksheet
Recursive Sequences Terry Anderson. What is a Recursive Sequence? A sequence that follows a pattern involving previous terms  To generate new terms,
Unit 5: Exponential Word Problems – Part 2
PreCalculus 5-R Unit 5 – Exponential and Logarithmic Functions.
Logarithmic, Exponential, and Other Transcendental Functions 5 Copyright © Cengage Learning. All rights reserved.
In case you don’t plan to read anything else in this powerpoint………. There is an activity you must do, somewhere hidden in this slide show, in preparation.
THE GOLDEN RATIO GREEK PRESENTATION 3 rd MEETING – BONEN –GERMANY October 2009 COMENIUS PARTNERSHIPS.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design Advanced Algorithms Analysis and Design By Dr. Nazir Ahmad Zafar.
Mathematical Connections.
The Fibonacci Sequence and The Goldens
7.4 Exploring recursive sequences fibonacci
A Number You Can Call Your Own
Chapter 3 Mathematics of Finance
The Golden Ratio and Fibonacci Numbers in Nature
Area Of Shapes. 8cm 2cm 5cm 3cm A1 A2 16m 12m 10m 12cm 7cm.
Area Of Shapes. 8cm 2cm 5cm 3cm A1 A2 16m 12m 10m 12cm 7cm.
1A Recursively Defined Functions
Linear, Quadratic, and Exponential Functions
Presentation transcript:

6.1 Golden Section 6.2 More about Exponential and Logarithmic Functions 6.3 Nine-Point Circle Contents 6 Further Applications (1)

6 Content P.2 A golden section is a certain length that is divided in such a way that the ratio of the longer part to the whole is the same as the ratio of the shorter part to the longer part. A. The Golden Ratio 6.1 Golden Section This specific ratio is called the golden ratio. Fig. 6.2

Further Applications (1) 6 Content P.3 Definition 6.1: 6.1 Golden Section Examples: (a) The largest pyramid in the world, Horizon of Khufu ( 柯孚王之墓 ), is a right pyramid with height 146 m and a square base of side 230 m. The ratio of its height to the side of its base is 146 : 230  1 : (b)Another famous pyramid, Horizon of Menkaure ( 高卡王之墓 ), is also a right pyramid with height 67 m and a square base of side 108 m. The ratio of its height to the side of its base is 67 : 108  1 : 1.61.

Further Applications (1) 6 Content P.4 Consider a line segment PQ with length (1 + x )cm. Divide the line segment into two parts such that PR = 1 cm and RQ = x cm. 6.1 Golden Section According to the definition of the golden section, we have Therefore, Fig. 6.5(a) Fig. 6.5(b)

Further Applications (1) 6 Content P Golden Section B. Applications of the Golden Ratio L 1 : W 1 is close to the golden ratio. (i) The Parthenon The Parthenon ( 巴特農神殿 ), which is situated in Athens ( 雅典 ), Greece, is one of the most famous ancient Greek temples. Fig. 6.8

Further Applications (1) 6 Content P Golden Section (ii) The Eiffel Tower The tower is 320 m high. The ratio of the portion below and above the second floor ( l 1 : l 2 as shown in Fig. 6.9) is equal to the golden ratio. Fig. 6.9

Further Applications (1) 6 Content P Golden Section C. Fibonacci Sequence The Fibonacci sequence is a special sequence that was discovered by a great Italian mathematician, Leonardo Fibonacci ( 斐波那契 ). This sequence was first derived from the trend of rabbits’ growth. Suppose a newborn pair or rabbits A 1 (male) and A 2 (female) are put in the wild. 1 st month : A 1 and A 2 are growing. 2 nd month : A 1 and A 2 are mating at the age of one month. Another pair of rabbits B 1 (male) and B 2 (female) are born at the end of this month. 3 rd month : A 1 and A 2 are mating, another pair of rabbits C 1 (male) and C 2 v (female) are born at the end of this month. B 1 and B 2 are growing. If the rabbits never die, and each female rabbits born a new pair of rabbits every month when she is two months old or elder, what happens later?

Further Applications (1) 6 Content P Golden Section Fig. 6.12

Further Applications (1) 6 Content P Golden Section Definition 6.2: The Fibonacci sequence is a sequence that satisfies the recurrence formula: According to the definition of the Fibonacci sequence, the first ten terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Further Applications (1) 6 Content P Golden Section Consider that seven squares with sides 1 cm, 1 cm, 2 cm, 3 cm, 5 cm, 8 cm, 13 cm respectively. Arrange the squares as in the following diagram: If we measure the dimensions of the rectangles, each successive rectangle has width and length that are consecutive terms in the Fibonacci sequence Then the ratio of the length to the width of the rectangle will tend to the golden ratio. Fig. 6.13

Further Applications (1) 6 Content P Golden Section D. Applications of the Fibonacci Sequence (a) In Music The piano keyboard of a scale of 13 keys as shown in Fig. 6.14, 8 of them are white in colour, while the other 5 of them are black in colour. The 5 black keys are further split into groups of 3 and 2. In musical compositions, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song. Note that the numbers 1,2,3,5,8,13 are consecutive terms of the Fibonacci sequence. Fig. 6.14

Further Applications (1) 6 Content P Golden Section (b) In Nature Number of petals in a flower is often one of the Fibonacci numbers such as 1, 3, 5, 8, 13 and 21.

Further Applications (1) 6 Content P More about Exponential and Logarithmic Functions Applications (a) In Economics Suppose we deposited $P in a savings account and the interest is paid k times a year with annual interest rate r%, then the total amount $A in the account at the end of t years can be calculated by the following formula In this case, the earned interest is deposited back in the account and also earns interests in the coming year, so we say that the account is earning compound interest.

Further Applications (1) 6 Content P More about Exponential and Logarithmic Functions (b) In Chemistry The concentration of the hydrogen ions is indirectly indicated by the pH scale, or hydrogen ion index. pH Value of a solution

Further Applications (1) 6 Content P More about Exponential and Logarithmic Functions (c) In Social Sciences Some social scientists claimed that human population grows exponentially. Suppose the population P of a city after n years obeys the exponential function where is the present population of the city. From the equation, the population of the city after five years will be approximately

Further Applications (1) 6 Content P More about Exponential and Logarithmic Functions (d) In Archaeology Scientists have determined the time taken for half of a given radioactive material to decompose. Such time is called the half-life of the material. We can estimate the age of an ancient object by measuring the amount of carbon-14 present in the object. Radioactive Decay Formula Where A 0 is the original amount of the radioactive material and h is its half-life. The amount A of radioactive material present in an object at a time t after it dies follows the formula:

Further Applications (1) 6 Content P Nine-point Circle Theorem 6.1: In a triangle, the feet of the three altitudes, the mid-points of the three sides and the mid-points of the segments from the three vertices to the orthocentre, all lie on the same circle. 6 Fig. 6.17