• Word count: 12487
• Level: International Baccalaureate
• Subject: Maths

#### A logistic model

IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg - Christian Jorgensen Creating a logistic model Christian Jorgensen IB Diploma Programme IB Mathematics HL Portfolio type 2 Candidate number International School of Helsingborg, Sweden IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg - Christian Jorgensen Theory A logistic model is expressed as: un?1 ? run {1} The growth factor r varies according to un. If r=1 then the population is stable. Solution . A hydroelectric project is expected to create a large lake into which some fish are to be placed. A biologist estimates that if 1x104 fish were introduced into the lake, the population of fish would increase by 50% in the first year, but the long-term sustainability limit would be about 6 x104. From the information above, write two ordered pairs in the form (u0, r0), (u0, r0) where Un=6x104. Hence, determine the slope and equation of the linear growth in terms of Un As Un =6 ?104 , the population in the lake is stable. Thus from the definition of a logistic model, r must equal to 1 as un approaches the limit. If the growth in population of fish (initially 1 ?104 ) is 50% during the first year, r must be equal to 1.5. Hence the ordered pairs are: (1?104 , 1.5) , (6 ? 104 , 1) One can graph the two ordered

• Word count: 8779
• Level: International Baccalaureate
• Subject: Maths

#### Creating a logistic model

Mathematics Higher Level Assignment: Mathematics Portfolio Type II: Creating a Logistic Model School: Trinity Grammar School School Code: Name of Student: Cassian Ho Student Number: Introduction If a hydroelectric project is expected to cerate a large lake into which some fish are to be placed, a biologist estimates that if 10,000 fish were introduced into the lake, the population of fish would increase by 50% in the first year, but the long-term sustainable limit would be about 60,000. In order to estimate the growth rate of the population of fish, it is best to find a linear growth factor for. We do this by finding two ordered pairs in the form (u0, r0), (un, rn). rn is the growth rate when the population is n. since we know from the information given that when there are 60000 fish in the lake, the growth rate is stable i.e. 1, this can be represented as one of our ordered pairs: (60000, 1). We also know that when 10000 fish are in the lake, the growth rate is 50% i.e. population is multiplied by 1.5, so the second ordered pair is (10000, 1.5). from these pairs we have: r10000 = 1.5 r60000 = 1 and since we are trying to search for a linear function of the growth rate, we can also denote rn as: rn = mn + b where n = population of fish. Substituting the two ordered pairs, we have: .5 = m(10000) + b = m(60000) + b Putting this into the GDC, we find

• Word count: 8216
• Level: International Baccalaureate
• Subject: Maths

#### Math Portfolio: trigonometry investigation (circle trig)

Math Honors 1: Trigonometry Investigation Task This investigation will present an analysis on initial problem by setting patterns and establishing mathematical relationships between the parameters in the problem. In this specific investigation, I will find to see the relationship between radius R and point X and Y in a coordinate plane. The center of the circle will be (0, 0) or the origin and the radius R will be unknown. Point P with the coordinate (x,y) will always be on the circumference of the circle, and will always be perpendicular from the X axis to the point. Part A: Circle Trigonometry . The diagram above, the radius r and the point (x, y) will form a right triangle. Therefore we can state that the equation to find the relationship can be described as Pythagorean Theorem. Wherever the coordinates (x, y) lies on the circle it will always form right triangle. Therefore we can use the equation distance between the two points. After we can state that the endpoints of the radius are set in the origin meaning that one end point will always set in the origin (0, 0). So we can simplify our equation further more. Then after squaring this will result in Pythagorean Theorem The diagram above with point R and Q is just to back up or example the idea of relationship use of right triangle to use the equation of Pythagorean Theorem. As a result, due to Pythagorean

• Word count: 7447
• Level: International Baccalaureate
• Subject: Maths

#### Math Studies I.A

Introduction This mathematics project will be examining many countries' life expectancy and GDP per capita and see which one is independent. Firstly, numbers that are of reliable resources from the internet will be found and will take the countries' life expectancy and GDP per capita (PPP). This experiment will be using correlation coefficient and the regression line to verify our results. Firstly, numbers will be taken from GDP per capita and overall life expectancy of both gender and using systemic sampling the data will be collected. The countries will only be collected at every odd My Null Hypothesis is that life expectancy is dependent to the country's GDP whereas my Alternative Hypothesis is that life expectancy and GDP are independent. The countries will be chosen in systematic sample of every odd number from a list of countries from The World Fact Book (see page:0 ). Then, the overall average of both sexes will be taken by adding both men and women's life expectancy and divided by 2, because I do not desire to see if GDP affects one of the sexes. Furthermore, the sex ratio of the total population will be looked at if there is much difference or not. Sex ratio is the ratio of males to females in a population. In addition, life expectancy only tells us the life expectancy of either male or female and by taking the average of both sexes one would have to take sex

• Word count: 6579
• Level: International Baccalaureate
• Subject: Maths

#### Lacsap's fractions - IB portfolio

Lacsap’s fractions MATHEMATICS SL INTERNAL ASSESSMENT TYPE 1 By: Veronika Kovács Lacsap’s fractions INTRODUCTION: The Lacsap’s fractions are a set of fractions arranged in a symmetrical patter. The task is to consider this set and to try to find a general statement for En(r) with (r+1)th element in the nth row, starting with r=0. To make the task easier ’ kx ’ will be the sign for the numerator of the xth row and ’mx’ will be the sign for the denominator of the xth row. 1 3/2 1 6/4 6/4 1 10/7 10/6 10/7 1 15/11 15/9 15/9 15/11 1 PATTERNS OF THE NUMERATOR: Graph 1 shows that the relation between n and kn is systematically rising: Graph 1: This graph shows the relation between n (the row number) and kn (the numerator of the nth row. Since I have found four patterns according to which the numerator can be calculated I have numbered each one to be able to distinguish them better. . A pattern of the numerators can be seen as they are arranged in a row where each term is one value greater than the absolute difference of the previous two numerators added to the previous term. kn kn-1 kn-2 2×kn-1-kn-2+1=kn 6 3 1 2×3-1+1=6 0 6 3 2×6-3+1=10 5 10 6 2×10-6+1=15 Table 1:

• Word count: 6298
• Level: International Baccalaureate
• Subject: Maths

#### Virus Modelling

IB HL Maths Modelling the Course of a Viral Illness and its Treatment Candidate Name: Sherul Mehta Centre Number:002144 Candidate Number: CSY 114 Contents Introduction - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg3 Modelling infection * Part 1. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg4 * Part 2. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg7 Modelling Recovery * Part 3. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg11 * Part 4. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg16 * Part 5. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg18 * Part 6. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg20 Analysing your models - - - - - - - - - - - - - - - - - - - - - - - -pg24 Applying your model * Part7. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg26 Introduction Through this coursework I will investigate the case of a standard adult patient who is infected by a virus and produce models to explain the entire process. I will look at the replication rates of the virus at different stages. Then I will show what happens when the immune response begins and how to eliminate the viral particles. Then I will show how and when medicine can be administered and I'll portray its effects on the

• Word count: 6140
• Level: International Baccalaureate
• Subject: Maths

#### Math Portfolio Type II

Mathematics Portfolio Type II Creating a Logistic Model Description A geometric population model takes the form where r is the growth factor and un is the population at year n. For example, if the population were to increase annually by 20%, the growth factor is r = 1.2, and this would lead to an exponential growth. If r = 1 the population is stable. A logistic model takes a similar form to the geometric, but the growth factor depends on the size of the population and is variable. The growth factor is often estimated as a linear function by taking estimates of the projected initial growth rate and the eventual population. Part 1 Information which has been given in Part 1: - a) 10,000 fishes are introduced into a lake b) The population increase if 10,000 fishes are introduced into the lake would be by 50% in the first year. c) The long term sustainable limit in this case would be 60,000 It has been given that the geometric population growth model takes the form . Now, if we have to find out the ordered pair (u0,r0): - * It is mentioned in the description that un is the population at year n. Therefore, u0 will be the population at year 0, or the initial population of the lake which is 10,000. Therefore, u0 = 10,000 ------------------- (1) * r is the growth factor as mentioned in the description. As the population would increase by 50% in the first year, so

• Word count: 5981
• Level: International Baccalaureate
• Subject: Maths

#### Stellar Numbers. In this task geometric shapes which lead to special numbers will be considered.

Maths Internal Assessment Type 1 - Mathematical Investigation Mathematics Standard Level Stellar Numbers September 2011 Luís Ferreira Teacher: Mr.Robson Aim - In this task geometric shapes which lead to special numbers will be considered. For example the easiest of these are square numbers which can be represented by squares of side 1, 2, 3 and 4. . The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers. Complete the triangular numbers sequence with three more terms. Find a general statement that represents the nth triangular number in terms of n.1 Finding a general statement: Now that three more terms have been drawn, the general statement can be found for the sequence; 1, 3, 6, 10, 15, 21, 28, 36. To do this, the constant difference in the sequence will need to be found, as shown below. This is needed to determine the type of equation (linear, quadratic, cubic etc...) 2 Number of term (n) 0 2 3 4 5 6 7 Sequence 3 6 0 5 21 28 36 First Difference 2 3 4 5 6 7 8 Second Difference 1 1 1 1 1 1 The standard rules to find the general statement were researched and the

• Word count: 5627
• Level: International Baccalaureate
• Subject: Maths

#### This essay will examine theoretical and experimental probability in relation to the Korean card game called Sut-Da. First, a definition of probability and how it is used in general life will be examined. Each hand of Sut-Da provides the theore

ABSTRACT This essay will examine theoretical and experimental probability in relation to the Korean card game called 'Sut-Da'. First, a definition of probability and how it is used in general life will be examined. Each hand of 'Sut-Da' provides the theoretical probability for a player to win the game. It is clear however, that the theoretical value of winning in 'Sut-Da' does not always apply in real life games. Secondly, the experimental probability of winning for each hand is examined. To find out the probability of winning with each hand, I am using permutation & combination, theoretical probability and experimental probability. Experimental probability data was gained from my friend and I playing the game. Finally, within my evaluation, I looked at experimental probability using excel spread sheets and also using calculations that were compared with the experimental probability data gained from actually playing the game. The theory of probability has been covered in a number of textbooks and I used these textbooks to help me get used to the formula. I have then worked out all the possibilities of hands and their probabilities for winning, performing all the calculations myself and using my own numbers in presenting my data for the experimental probability data. Introduction It all began when I started to watch a Korean drama called "Ta-JJa". I decided to watch this

• Word count: 5446
• Level: International Baccalaureate
• Subject: Maths