QUEENSLAND ACADEMY OF SCIENCE MATHS AND TECHNOLOGY Mathematics IA SL Type 2 Jeremiah Joseph 0/8/2009 This Internal Assessment will investigate models of female BMI data. BMI is the measure of one's weight to their height. To calculate a person's BMI their weight is divided by the square of their height. Shown below is data for female BMI values in the US in 2000. Age (years) BMI 2 6.40 3 5.70 4 5.30 5 5.20 6 5.21 7 5.40 8 5.80 9 6.30 0 6.80 1 7.50 2 8.18 3 8.70 4 9.36 5 9.88 6 20.40 7 20.85 8 21.22 9 21.60 20 21.65 The data points shown above were graphed, the resulting graph is shown below. The variables that were used in the graph above were age and BMI. The independent variable, age, was placed on the x axis. Age is the independent variable because it is constant. The dependant variable, BMI, was placed on the y axis. BMI is the dependant variable because is varies, dependant on the age. It is clearly shown in the graph above that the BMI of females in the US in 200 can be modelled using the equation y= A sin (Bx-c) + D. This is because the graph is shown to have the same characteristics of a sin graph. In this equation A is the amplitude of the graph. Where max = maximum dependent variable value and min = minimum dependent variable value. The maximum value obtained from the data is 21.65 whereas the minimum value is 15.20.
Problem Statement: The game of Unlucky 7 is one of the most famous (or notorious) of all gambling games played with dice. In this game, the player rolls a pair of six-sided dice (assumed to be fair), and the SUM of the numbers that turn up on the two faces is noted. If the sum is 7 or 11 on the first roll, then the player wins immediately. If the sum is 2, 3, or 12 on the first roll, then the player loses immediately. If any other sum appears, then the player continues to throw the dice until the first sum is repeated (the player wins) or the sum of 7 appears (the player loses). Requirements: . Compute the theoretical probabilities for each sum of a dice roll (rounded to the nearest thousandth decimal place). Use for the simulation model. o Since the random variable is the sum of the two fair die, we need to create a probability table that correlates to the probability of the sums 2-12, inclusive. o From prior knowledge, I know that this particular probability distribution is binomial, and the total number of ways is 6² because there is six possibilities on a fair dice, and since we are rolling two at the same time, it is the exponent. RV = SUM P(SUM) Decimal Approximation 2 /36 .028 3 2/36 .056 4 3/36 .083 5 4/36 .111 6 5/36 .139 7 6/36 .166 8 5/36 .139 9 4/36 .111 0 3/36 .083 1 2/36 .056 2 /36 .028 o The decimal approximation for
Portfolio Type 1 investigation Transformation of Trigonometric Functions Investigate the function: y = a sin b(x-c) + d in respect to the transformation of the base curve of y = sin x, depending on the values of a, b, c, and d. Be sure to consider all possible values for a, b, c, and d. Describe the base curve Start with a (try some values) Then try values in b (sin2x, sin -3x, sin1/2x) Does your hypothesis hold true for y = cos(x) and y = a cos b (x-c) +d? How about tan(x)? Transformation of Trigonometric Functions Introduction The purpose of this study is to examine the transformation of trigonometric functions of y=A sin B (x-C) + D and determine the effect on the base curve y=sin(x). I am going to be systematically changing the values of A, B, C and D in the equation y= A sin B (x-C) +D. First I am going to examine different numbers for the value of A. I am going to use whole numbers, negative whole numbers, positive rational numbers and negative rational numbers for the value of A and see how this affects the Sine curve. Then I will examine different numbers for B, then C then D. After examining the Sine function I test to see if changing the values of A, B, C and D will have the same effect on the Cosine function. Sine Curve: Figure 1 Figure 2 I will use y-sin (x) as my base curve. This has a maximum value of 1, minimum value of -1
Mathematics SL Parellels and Parallelograms. This task will consider the number of parallelograms formed by intersecting m horizontal parallel lines with n parallel transversals; we are to deduce a formula that will satisfy the above.
Parallels and Parallelograms Mathematics Coursework This task will consider the number of parallelograms formed by intersecting m horizontal parallel lines with n parallel transversals; we are to deduce a formula that will satisfy the above. Methodology . We started out the investigation with a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed (shown in Figure 1) 2. A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are formed: A1, A2, and A1?A2 3. When a fourth transversal is added to Figure 2 (Figure 3), six parallelograms are formed. A1, A2, A3, A1?A2, A2?A3, A1?A3 4. Figure 4 has 5 transversals cutting the pair of horizontal parallels, forming ten parallelograms. A1, A2, A3, A4, A1?A2, A2?A3, A3?A4, A1?A3, A2?A4, A1?A4 5. A sixth transversal was added to Figure 5, forming 15 parallelograms shown in Figure 6. A1, A2, A3, A4, A5, A1?A2, A2?A3, A3?A4, A4?A5, A1?A3, A2?A4, A3?A5, A1?A4, A2?A5, A1?A5 6. When a seventh transversal is added, twenty-one parallelograms are formed (Figure 7). A1, A2, A3, A4, A5, A6, A1?A2, A2?A3, A3?A4, A4?A5, A5?A6, A1?A3, A2?A4, A3?A5, A4?A6, A1?A4, A2?A5, A3?A6, A1?A5, A2?A6, A1?A6. 7. I then used technology (table 1.0) to record the above and calculate the differences between the parallelograms formed with each addition of a
Type I - Parallels and Parallelograms
Parallels and Parallelograms Math Portfolio Introduction: 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. These lines represent two transversals; however they are supposed to intersect with a horizontal parallel line. These lines represent two horizontal parallel lines. They are intersected with a number of transversals. When there is one horizontal line and two transversals, this makes one parallelogram, A1, as shown below: This diagram shows that when there are two horizontal lines and two transversals, one parallelogram is formed. This parallelogram is called A1. However, when another transversal is added to the same diagram and same pair of horizontal lines, it looks like this: This figure shows that when another transversal is added to a pair of horizontal parallel lines, three parallelograms are formed. Although the third parallelogram is not labeled, it is clear that the total of A1 A2 makes a big parallelogram, A3. So therefore, there is A1, A2, and A3. Another transversal added to this
portfolio Braking distance of cars
. Use a GDC or graphing software to create data plot of speed versus thinking distance and data plot of speed versus braking distance. Describe your results. The data plot of speed versus thinking distance. The speed is relative to the thinking distance. So, this graph shows the direct proportion. There is no way that these values are minus because it doesn't make sense that thinking distance is minus. In this report, let the time of stepping on the brake is equal. The data plot of speed versus braking distance. The graph should show exponential function. It is conceivable that it causes friction when the car breaks. When the car starts breaking, the speed of car is on the table below. During the car keep breaking, speed is decreasing due to friction. Speed (km/h) Speed(m/s) 0 0 32 8.888 48 3.33 64 7.77 80 22.22 96 26.66 12 31.11 The relationship between the speed and advanced distance is shown the graph below. x = speed (m/s) y=advanced distance First, the speed of car is constant. Speed will be decreasing according to braking cause friction. Thus the advanced distance will be decreasing as well. At least, the car will stop which indicate point 0. Therefore, area of triangle represents braking distance (BD). Speed and distance are not constant because friction affects these aspects. BD = = = That's why the relationship between speed and
Matrix Binomials IA
Maths Portfolio Standard Level International Baccalaureate Matrix Binomials The main aim of this portfolio is to investigate the matrix binomials and observe and determine a general expression from the patterns that we obtain through the workings. Throughout the project, I shall be using solely matrices of 2 x 2 formations, and investigate the patterns I find. . To begin with, we consider the matrices X = and Y =. The values of these matrices, each raised to the power of 2, 3 and 4 are calculated, as shown below; X2 = X = Y2 = x = X3 = x = and Y3 = x = X4 = x = Y4 = x = It can be observed that all the matrices calculated above are in the form of 2 X 2, they are all square matrices. The corresponding diagonal elements are also observed to be the same. Since the matrices of each nth power can be seen to be the value of 1 less than the nth term, the general expression for the matrix Xn in terms of n is - Xn = And the general expression for Yn is - Yn = Likewise, the values of the matrix (X + Y), raised to the power 2, 3, and 4 is calculated to find its general expression. The matrix: (X + Y) = + = So, (X + Y) 2 = = (X + Y) 3 = x = (X + Y) 4 = x = And from the above, we can infer that the general expression for (X + Y) n is as follows, (X + Y) n = Proof: Taking n as 3, the value is
models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000
Anh Nhu Vu IB Mathemathics Standard Level 2008 Maths Coursework This coursework will explore models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000. The ages and the corresponding BMI numbers are variables and as we generate the function that models the behaviour of the graph later on, the parameters are the values of a, b, c and d in the formula of the function. When the data points are plotted on a graph, it is interesting to see what kind of graph these points are forming. The graph BMI appears to resemble the graph of a trigonometric function. If we base on the function y=sinx to develop our model function, the type of function that models the behaviour of the graph is: y1=a×sin [b(x+c)] +d These are the reasons why I chose this type of function: Primarily, the shape of the graph resembles that of the graph of the function y=sinx. However, in comparison to the graph of the function y=sinx, the data points of the graph BMI indicates that the processes of transformation have been done: _Vertical and horizontal stretch (the scales are represented by a and b) _Vertical and horizontal translation (the scales are represented by c and d) Once we have indentified the type of the function that models the behaviour of the graph, it is possible to use algebraic methods to create an equation that
The Two Yachts Problem
The Two Yachts Problem Pg. 405 IB Math SL Y2 Yacht A has initial position (-10, 4) and has velocity vector. Yacht B has initial position (3, -13) and has velocity vector. . Explain why the position of each yacht at time t is given by rA = + t and rB = + t - For vector equations, the form is =. ( refers to an initial position and refers to a direction vector.) - Therefore, a vector equation for Yacht A can be written as + t. - A vector equation for Yacht B can be written as + t. 3. The position of B relative to A () is rB - rA = + t, which in coordinates will be. 4. The formula for finding the distance is: d = Therefore, d2 = 169 - 78t + 9t2 + 289 - 136t + 16t2 = 25t2 - 214t + 458 5. d2 is a minimum when t = 4.28 To find the minimum of d2, I set the derivative equal to 0. So, 50t - 214 = 0. Thus t = = 4.28. 6. The time when d is to be a minimum is the same time as when d2 is a minimum, so the closest approach occurs at t = 4.28. So, if I put t = 4.28 into the expression for d is: d = = = = = 0.2 miles
Plot a graph for the BMI of different females of different ages in the US in year 2000 and analyse whether it is an accurate source of data.
Body Mass Index Maths Coursework March 2008 By: 12M(2) This maths coursework is based on Body Mass Index. This is a measure of ones body fat; it is calculated by taking one's weight (kg) and dividing it by the square of one's height (m). For this coursework, I have to plot a graph for the BMI of different females of different ages in the US in year 2000 and analyse whether it is an accurate source of data and how it can help me to find other BMI's around the world. Below is my graph showing the BMI of different females of different ages in the US in the year 2000: As shown on the graph, the 'x' values are the age of the females and the 'y' values are their body mass index. The age is measured in Years. When modelling this data, the initial impression is to think that is was an f(x)= x2 graph. However once you notice that it is not a mere parabola but a wave due to the curve that levels off (shown on graph) we can assume it is a periodic function such as a cosine or sine graph. Even though you can use a cosine or a sine graph, I decided to use a cosine graph, as I am more familiar with this type of graph. After inputting the cosine function into autograph software, you would realise that transforming the function would be appropriate so that it can model the graph more accurately. In order for you to do this, you have to use the general formula of f(x)= acos(bx