#### Mathematics (EE): Alhazen's Problem

Extended Essay - Mathematics Alhazen's Billiard Problem Antwerp International School May 2007 Word Count: 3017 Abstract The research question of this Mathematics Extended Essay is, "on a circular table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge". In investigating this question, I first used my own initial approach (which involved measuring various chord lengths), followed by looking at a number of special cases scenarios (for example when both balls are on the diameter, or equidistant from the center) and finally forming a general solution based on coordinate geometry and trigonometric principles. The investigation included using an idea provided by Heinrich Dorrie and with the use of diagrams and a lengthy mathematical analysis with a large emphasis on trigonometric identities, a solution was found. The conclusion reached is, "if we are given the coordinate plane positions of billiard ball A with coordinates (xA, yA) and billiard ball B with coordinates (xB, yB), and also the radius of the circle, the solution points are at any of the points of intersection of the circular table with the hyperbola,", where P =, M = , p = , m = and r is the radius. The solution was verified by considering specific examples through technology such as Autograph software and a TI-84

#### Investigating the Graphs of Sine Functions.

03.02.2004 Anna Markmann Portfolio #2 - Type I Investigating the Graphs of Sine Functions The purpose of this assignment is to obtain general rules for transformations of sine functions from analysing patterns got from examples of these. To justify my conjectures of all of the following functions I used "Magic Graph" - an electronic graphing program which allowed me to present them with a high level of precision. The trigonometric settings and the radian mode were kept constant throughout the whole investigation. Part 1 Graph of y= sin x To present this graph properly there are several possibilities: one can use a graphing calculator, a computer program, draw the graph from tabled values or from the unit circle. I chose the unit circle method because it is then more understandable how sine of x gets its shape and position, since sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse and therefore1: Characteristics of y= sin x The sine curve is symmetric with the origin, it is an even function, and has infinite intercepts at multiples of , as well as infinite maximum and minimum points at -1 and 1. Because the coefficient of sine is 1, the amplitude of y= sin x, the distance from the central value or the height of each peak above the baseline, is 1. As the period of a function is the length of the time the system takes to

#### Investigating Divisibility

. Factorise the expression P(n) = nx - n for x ? {2, 3, 4, 5}. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your results by showing whether P(k+1) - P(k) is always divisible by x. Using appropriate technology, explore more cases, summarize your results and make a conjecture for when nx - n is divisible by x. .1 Factorise the expression P(n) = nx - n for x ? {2, 3, 4, 5}. When n is real numbers; x = 2 P(n) = n2 - n = n (n - 1) x = 3 P(n) = n3 - n = n (n2 - 1) = n (n + 1) (n - 1) x = 4 P(n) = n4 - n = n (n3 - 1) = n (n - 1) (n2 + n + 1) x = 5 P(n) = n5 - n = n (n4 - 1) = n (n2 + 1) (n2 - 1) = n (n2 + 1) (n - 1) (n + 1) .2 Determine if the expression is always divisible by the corresponding x. 1.2.1 x = 2 P(n) = n2 - n = n (n - 1) divisible by 2? When n = 1 n (n - 1) = 1 (1 - 1) = 1 (0) As 0÷2 = 0, P(n) is divisible by 2 when x = 2 and n = 1 Assume n = k is correct k (k - 1) = k2 - k = 2M (where M is any natural number) Then considering n = k + 1 (k + 1) (k) = k2 + k = (k2 - k) +2k = 2M + 2k = 2(M + k) Therefore, P(n) = n2 - n is divisible by 2 when x = 2 and n = any natural numbers. 1.2.2 x=3 P(n) = n (n + 1) (n - 1) divisible by 3? When n = 1 n (n + 1) (n - 1) = 1 (2) (0) = 0 As 0÷3 = 0, P(n) is divisible by 3 when x = 3 and n = 1 Assume n = k is correct k (k + 1) (k - 1)

#### Barbara & Allen's Compound Interest

Angela C. Rosario Mr. Thomas IB Mathematics SL I (3) 24 March 2009 Practice IB Internal Assessment Introduction: The purpose of interest is for a bank to pay an individual for the use of their money. Interest therefore represents one's return on the investment. To calculate n interest compoundings per year, one must utilize the formula: . Alan invests $1000 at an interest rate of 12% per year. Copy and complete Table 1 which shows A, the value of the investment in dollars after t years, assuming that the interest is compounded yearly. One must utilize the formula for Compound Interest in order to determine the answer. In the case of Alan, his principle (P) amount of money is $1000, and he collects interest at a rate (r) of 12% per year. To determine the value of the investment in dollars after t years in order to satisfy Table 1, the formula necessary is: A= 1000 ( 1+ .12/1)(1)(t) where P= $1000, r= 0.12, and n=1 because interest is being compounded annually or once a year, and t= the number of years the money is present in the account. t=0, A= 1000( 1+ 0.12/1)(1)(0) = 1000(1+ 0.12) 0= 1000(1.12) 0= 1000(1)= 1000 t=1, A= 1000( 1+ 0.12/1)(1)(1) = 1000(1+ 0.12) 1= 1000(1.12) 1= 1000(1.12)= 1120 t=2, A= 1000( 1+ 0.12/1)(1)(2) = 1000(1+ 0.12) 2= 1000(1.12) 2= 1000(1.2544)= 1254.40 t=5, A= 1000( 1+ 0.12/1)(1)(5) = 1000(1+ 0.12) 5= 1000(1.12) 5˜ 1000(1.76234)˜

#### Infinite Summation- The Aim of this task is to investigate the sum of infinite sequences tn.

Infinite Summation Math Portfolio Introduction: Aim of this task is to investigate the sum of infinite sequences tn. An infinite sequence is a listing of numbers with no limits, like: (1;2;3;4;5;…) Whereas the three dots (…) at the end of the sequence states the infinity. The sum of an infinite sequence can be determined, depending on the sequence. Sequences are categorised in geometric sequences and arithmetic sequences, where in both notation and calculations differ from each other. An arithmetic sequence is a series of numbers, where the common difference (d) between terms is constant; the common difference is added or subtracted to the term before. A geometric sequence on the other hand, has a common ratio (r), where r will be multiplied to the term before, which also is constant. For both types of sequences, it is possible to calculate specific terms and the sum to certain point, whereas the number of terms is given by (n). In this task the infinite sequence of tn, will be examined, where t0=1, t1=(xlna)1,t2=(xlna)22×1,t3=(xlna)33×2×1 …,tn=(xlna)nn!… As the general sequence states n!, which is factorial notation and is used to simplify. Factorial notation (n!), is defined by n!=nn-1n-2…3×2×1 Note that 0!=1 To observe, how the sum of the sequence changes, it will be assumed that x=1 and a=2, , (ln2)1,(ln2)22×1,(ln2)33×2×1… The

#### Genetic algorithm

Department of Computer Engineering Subject: Applied Information Theory Genetic algorithm Done by: Dinara Sabazova Gaukhar Zharylgapkyzy 2nd year student FIT, AC Checked by: Kaimov A. Senior lecturer Almaty 2013 I. Introduction First Words Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. As you can guess, genetic algorithms are inspired by Darwin's theory about evolution. Simply said, solution to a problem solved by genetic algorithms is evolved. History Idea of evolutionary computing was introduced in the 1960s by I. Rechenberg in his work "Evolution strategies" (Evolutionsstrategie in original). His idea was then developed by other researchers. Genetic Algorithms (GAs) were invented by John Holland and developed by him and his students and colleagues. This lead to Holland's book "Adaption in Natural and Artificial Systems" published in 1975. In 1992 John Koza has used genetic algorithm to evolve programs to perform certain tasks. He called his method "genetic programming" (GP). LISP programs were used, because programs in this language can express in the form of a "parse tree", which is the object the GA works on. II. Search Space Search Space If we are solving some problem, we are usually looking for some solution, which will be the best among others. The space of all feasible solutions

#### Music and Maths Investigation. Sine waves and harmony on the piano.

Transfer-Encoding: chunked Page EXPLORATION OF THE RELATIONSHIP BETWEEN MATHEMATICS AND MUSIC ________________ INTRODUCTION I have chosen to investigate the mathematical properties of music; to be more precise, the piano. Since an early age I had a great interest and appreciation for music, in addition, I had decided to play the piano and now I have been playing it for 11 years. I enjoy listening to various types of composers and musical artists varying from classical pieces to modern pieces. However, during my last year in my piano classes, I’ve got a spark of interest in figuring out how composers manage to compose pieces that are appealing to me and the public, especially, when some composers had health problems, which made the journey of creation quite hard for them, in this case, Beethoven created wonderful musical pieces while being deaf. After some personal research, I’ve found out that mathematics has a huge role in determining how it appeals to the public, as mathematics is related to the production of harmony. Thus, the aim of my exploration is to explore and understand how mathematics are related to harmony of music. All of the aspects of mathematics in relation to harmony will be focused on my main instrument – the classical piano. Although music is made out of all of the aspects such as melody, harmony and rhythm, however, harmony is very intricate

#### The speed of Ada and Fay

IB Student DP1 HL Mathematics Internal Assessment Type II My Internal Assessment In this internal assessment, I am going to create a recursive formula for a mathematic model. In this investigation, I use real life examples and uses real data to finish my investigation. Afterward, I place my model to real life case study. However, there are different limitations in different cases. Some of the limitations are dimensional plane, speed, boundaries and more. At last, I am going to present my recursive formula, . The speed of Ada and Fay After reading the Internal Assessment question sheet, I had an idea on using a similar real life case study. Then, I invite my cousin, Ada and her dog, Fay, to have a running test and let me collect a set of data for my Internal Assessment. Therefore, I set a straight track with a distance of 100 meters and they need to run for 10 times. So, I can collect a set of accurate data. After each of them run for 10 times, I will take an average time for the 100 meters run and then uses the speed formula, . In this formula, V stands for velocity, which is the speed; the S stands for distance, at last, T stands for the period of time the runner runs. Below is Ada's data for running 100 meters: Ada's running records Trails Time (s) 6.6 2 6.7 3 6.8 4 6.7 5 6.6 6 6.5 7 6.6 8 6.8 9 6.5 0 6.7 After collecting Ada's set of data, then

#### Math Portfolio Type 1

Math Portfolio Type 1 Task Investigating areas and volumes Miras International School Amrebayeva Nurbala Gr. 12IB Introduction In this portfolio, I'm going to investigate ratios of areas and volumes of power functions, which can be generalized as the following: Y=xn, where n-is the power, n?R This function will be graphed between two arbitrary parameters x=a and x=b such that a<b (in other words boundaries or limits). In this investigation, I'm going to use method of math induction, integration using power rule, application of integration (areas under the curve and solids of revolution), also some knowledge about power functions. Investigation Process Given the power function y=x2, graph of which is parabola. I need to consider the region formed by this function from x=0 to x=1 and the x-axis, let's label this area B; and the region from y=0 to y=1 and the y-axis area A. This can be shown on the illustration (Fig.1) below: Figure 1 To plot this, I used Graph 4-3 Software, which I found very convenient. So the areas formed and illustrated above (Figure 1), need to be considered in order to find a ratio between them, hence area A: area B. To find that, I need to use integration using power rule. Integration, using power rule is the general integration rule, which is: where n?-1 It consists of the following steps: . Calculate area B, which is -0 2. Calculate

#### Modelling Probabilities in Games of Tennis

Math HL Portfolio Type II: Modelling Probabilities in Games of Tennis May 2009 In this portfolio we will look at the probability involved in playing tennis. Our calculations will be based on the estimated probability a player has of scoring a point. We will develop models for different kinds of tennis games and use Excel to explore up to what extent we can exploit the two probabilities with which we start. Furthermore, we will differentiate between probability and odds, comparing them and analyzing how they can affect the way we look at the same numbers. In my conclusion I will mention the possibility of involving other kinds of distribution in this portfolio, such as Poisson. Part 1: Club Practice. . Games to 10 points. a) Since we know that Adam wins about twice as many points as Ben does, we can say that the probability of Adam winning a point is, and the probability of Ben winning a point is. So, given that P(A) is the probability of Adam winning a point and P(B) is the probability of Ben winning a point, we have that: P(A)= P(B)= Clearly, this is a binomial distribution. Hence, we will use the formula, where n is the total number of trials, x the number of successes, p the probability of success and q the probability of failure. Because we want the variable x to represent the number of points won my Adam, we will substitute P(A) for p and P(B) for