#### Math IA type 2. In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis.

Nikhil Lodha Mathematics Internal Assessment Type 2 Mathematical Modelling Modelling Probabilities in games of tennis. IB Math HL Mr. Hoggatt 2nd June 2009 Introduction In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis. I will look to start with a relatively easy and simplistic models where Adam and Ben play each other in Club practice and have a set number of point that they will play. I will then look to find an expected value for the number of points that Adam wins. For this expected value I will calculate a standard deviation to see how much does a randomly selected point vary from the mean. I will then look at Non Extended play games where a maximum of 7 points can be played. I will show that there are 70 ways in which the game can be played. I will do this with the help of the binomial probability distribution formula. I will also calculate the odds of Adam winning the game and then look to generalize my model so that it does not only apply to only Adam and Ben but to any player. After making generalized model, I will look at extended games where in theory games could go on forever. Here I will look to use the sum of an infinite geometric series to come up with an appropriate model. I will then use that model to find the odds of Adam winning extended games and then I will look to

#### Math Portfolio Type II Gold Medal heights

The Olympic games were created thousands of years ago with the purpose to determine who could push himself the hardest to jump higher or run faster. High jump has always been part of the revived Olympics and has been carried out ever since the 1896's Olympics. The achievements of athletes almost put itself forward for analysis, as the data is already precisely recorded and therefore it is rather easy to make predictions. This data is useful for trainer or scouts when they have to estimate whether an athlete is worth their while or not. As in all of the other programs a steady improvement of the athletes' performances have been recorded as the games advanced and almost all skilled sportsmen were taken under supervision of trainers and were made professionals. The recorded heights of the gold medal winners starting from the 1932's Olympics until 1980 are as follow: Year 932 936 948 952 956 960 964 968 972 976 980 Height in cm 97 203 98 204 212 216 218 224 223 225 236 Table 1.0 Data table for the height of the gold medallists in high jump from 1932 until 1980. Using these raw values to plot a graph without any manipulation results in a picture which is both hard to read and not well suited to convey the wanted information. If the x-Axis was to correspond with the years starting from year zero, the graph would be unnecessarily broad. The data of

#### IB Math Methods SL: Internal Assessment on Gold Medal Heights

International Baccalaureate IB Mathematics SL Y2 Internal Assessment Task: Gold Medal Heights (Type 2) Introduction The aim of this investigation is to consider the winning height for the men’s high jump in the Olympic Games. Firstly, we are given a table that lists the record height achieved by gold medalists in each competition from 1932 onto 1980. Given Information Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 Height (cm) 197 203 198 204 212 216 218 224 223 225 236 N.B. The Olympic games were not held in 1940 and 1944. Let us first plot the points on a graphing application. After plotting the above points in the graphing programme (see Figure 1 in Appendix), we arrive at the graph below. Graph 1: Height achieved by gold medalists in various Olympic Games Domain: 1920 t 1990 Range: 195 t 240 This graph demonstrates the record height changing over time through every Olympic game. The h-axis represents the height of the record jump (in centimeters). The t-axis represents the years in which the jumps took place. It should be noted that there are certain limitations to this data set; primarily that there was no men’s high jump taking place in 1940 and 1944; none in 1940 due to the revocation of Tokyo as the host venue for the Games due to the Sino-Japanese war; and none in 1944 due to the outbreak of the Second

#### Math portfolio: Modeling a functional building The task is to design a roof structure for the given building. The building has a rectangular base 150 meters long and 72 meters wide. The height of the building should not exceed 75% of its width

Math portfolio: Modeling a functional building The task is to design a roof structure for the given building. The building has a rectangular base 150 meters long and 72 meters wide. The height of the building should not exceed 75% of its width for stability or be less than half the width for aesthetic purpose. The minimum height of a room in a public building is 2.5 meters. The height of the structure ranges from 36m to 54 (72×75%) m as per the specification. At first I will model a curved roof structure using the minimum height of the structure that is 36 meters. From the diagram given, the curve roof structure seems to be a parabola hence I will use a general equation of parabola that is y= ax2 + bx +c -----------(1) Now the width of the structure is 72 meters and the height is 36 meters. Let the coordinate of the left bottom corner of the base is (0,0) Then the coordinate of the right bottom corner will be (72,0) and the coordinate of the vertex of the parabola will be (36,36) Since the above three points lies in the parabola, we will get 3 equation by substituting these coordinate in equation (1) C=0 -------(2) 5184a + 72b = 0 -------(3) 296a + 36b = 36 ------------(4) Solving: 5184a + 72b = 0 5184a =-72b a = a = Substituting a = to (4) 296( ) + 36b = 36 -18b + 36b = 36 b = 2 Therefore: a = , b = 2 , c = 0 So the equation for the curved roof

#### Mathematic SL IA -Gold medal height (scored 16 out of 20)

Mathematics Standard Level Portfolio Type 2- gold medal height Candidate name: Sun Ha (Rucia). Park Candidate number: School: Beijing No.55 High School ________________ Content Introduction ________________ Introduction Olympic Game is a variety competition of various athletics. It is composed twice a year, summer and winter, and it is hold once four year. Because there are many athletics competitions, it’s a great opportunity to take numerical data. In this task, I will investigate the winning height for the men’s high jump in the Olympic Games. The high jump is a track and field athletics plays in which athletes jump over a horizontal bar placed at measured heights without any aid of certain devices. I m going to use Microsoft Excel and the graphing calculator TI-84 to collect and present the data for analysis and investigation, and find out the best –fit functional curve to the original data. The table has given the original numerical data of the height achieved by the gold medalists at various Olympic Games. Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 Height (cm) 197 203 198 204 212 216 218 224 223 225 236 Depending on the data, I will investigate this case by setting time of Olympic for my independent variable, and setting the winning height for my dependent variable. In graphs I will show, I let x denote

#### Fishing Rods

Year 11 IB Maths - Portfolio Type II Fishing Rods A fishing rod requires guides for the line so that it does not tangle and so that the line casts easily and efficiently. In this investigation, a mathematical model will be developed using matrix methods, polynomial functions, and technology to calculate functions from the given data points of two fishing rods of lengths 230cm and 300cm. This model will further determine the placement of the line guides on the fishing rod. In mathematics, a function is a relation between a given set of elements and another set of elements that associate with each other, algebraically and graphically. In this investigation, an approach using matrices will be attempted to calculate functions for the given data and then be plotted to verify the results. Furthermore, there will be an effective use of technology, using Graphic calculator and excel, so as to minimize errors and flaws. The first investigation is of Leo's fishing rod: Leo has a fishing Rod with overall length 230cm. The table below gives the distance for each of the line guides from the tip of the fishing rod. Guide number (from tip) 2 3 4 5 6 7 8 Distance from tip (cm) 0 23 38 55 74 96 20 49 Firstly, before a mathematical model can be formulated, we must outline and define the variables and constraints associated with the values given above. Independent

#### The Population of Japan and Swaziland

The Population of Japan and Swaziland Type 2 Portfolio Rachel Timmons Lee's Summit West High School IB Mathematics SL 5/18/2009 Population models are formulas that one can use to calculate the future population of a country based on past growth. These growths can sometimes be shown exponentially. In this portfolio, I will be finding population models for the countries of Swaziland and Japan. I will begin with Swaziland. Using the data in the following table, I will find an exponential function algebraically to describe the population based on the year. A possible format for this function is, and this is the one I'll be basing my model on. The following data was taken from www.library.uu.nl/wesp/populstat/afica/swazilac.htm . The populations shown are estimates. Year Population (thousand) Year Population (thousand) 911 00.0 960 330.0 921 12.8 970 422.0 927 22.0 980 565.0 936 56.7 990 751.0 944 71.3 2000 083.3 950 264.0 2005 317.0 In order to make the data easier to work with, I'm going simplify the years according to 1911 being year 1.Therefore, year 1921 will be represented as year 11, 1927 as 17, 1936 as 26, etc. These new expressive values will be used as my x values and the population as my y values. I placed these values in the table below. X Y X Y 00.0 50 330.0 1 12.8 60 422.0 7 22.0 70 565.0 26 56.7 80

#### Parallels and Parallelograms Maths Investigation.

CAYMAN INTERNATIONAL SCHOOL PARALLELS AND PARALLELOGRAMS JULIANA N. WOOD 9/7/2012 ________________ Definitions: Transversal: A line that cuts across a set of lines or the sides of a plane figure. Transversals often cut across parallel lines. Parallel line: Two distinct coplanar lines that do not intersect. Note: Parallel lines have the same slope. Parallelogram: A quadrilateral with two pairs of parallel sides. A1 : ᴗ : This investigation aims at finding a relationship between the numbers of horizontal parallel lines and the transversals. When these lines intersect they form parallelograms. The aim of this investigation is examine and determine a general statement for transversals and horizontal lines and how they affect the number of parallelograms formed within the figure. A diagram of a parallelogram and a transversal is shown below. Figure 1: two transversals Figure 1 below shows a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed. A1 Adding a third transversal A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are formed: A1 , A2 , and A1 ᴗ A2. A2 A1 Figure 2: three transversals Adding a third transversal gives us a total of three parallelograms. Adding a fourth transversal Figure 3: four transversals A1 , A2 , A3, A1 ᴗ A2, A2 ᴗ A3 and A1

#### In this investigation, I will be modeling the revenue (income) that a firm can expect given it demand curve using my knowledge of linear and quadratic functions

Demand & Revenue Investigation Grade 11 Maths Standard Level Reece Chau 11DZBH ________________ Grade 11 Mathematics SL Wednesday 2nd November 2011. Demand & Revenue Investigation. Aim: In this investigation, I will be modeling the revenue (income) that a firm can expect given it demand curve using my knowledge of linear and quadratic functions. Background Information: The firm we are focusing on is the Very Big Gas Company (VBGC). The VBGC is a government monopoly that supplies natural gas to a national market. The vast majority of its sales involve selling natural gas to consumers who use it for heating of homes and businesses. Market studies have shown that the demand for its product varies each quarter according to seasonal temperatures. Since VBGC is a national monopoly, the price of its product is regulated by a government agency so as to protect consumers from excessively high prices and to maintain a level of consumption that reflects national environmental goals. For each quarter, data on price and quantity sold has been collected and is recorded in a table below. The price is measured in Euros per cubic metre and the quantity is measured in millions of cubic metres of natural gas. Table showing the price and quantity of gas sold for each quarter. Quarter 1 Jan,

#### Investigating Slopes Assessment

Investigating Slopes Alessandro Marazzani 25/11/13 ________________ Maths Assessment Introduction The aim of our assessment is to find general laws and correlations with the investigation of slopes. In fact, we had been given a sheet, where we had many points to analyse and check and, at the end, come up with a conclusion. We are allowed to use our graphic calculator, mainly because we need to show our results. I will structure this essay by firstly doing an introduction, and then I will start by answering each point we have been given to analyse and come up with a statement at the end of each point. At the end, I will write a final conclusion that sums up all the results I found in this big investigation of slopes. Investigation 1 The first investigation of this assessment tells me to find a formula for the gradient when there is the function f(x)= X2. It also tells me to write the answer in the form of f1(x). So, to do this, I will plot the function in the graph, check the gradient, and see if there is a formula for it. This investigation is quick to complete, so I recon, I don’t need to do anything to my results, but only find a connection between them. f(x)= X2 X=? Tangent Gradient X= -2 Y= -4x+(-4) -4 X= -1 Y= -2x+(-1) -2 X= 0 Y= 0 0 X= 1 Y= 2x+(-1) 2 X= 2 Y= 4x+(-4) 4 The conclusion I can draw out from this investigation is that at