#### mathsbreaking

Mathematics Course Work Dominique Albert-Weiss "Stopping disntances" Table 1 Speed v (km/h) Thinking distance (m) Braking distance (m) 32 6 6 48 9 4 64 2 24 80 5 38 96 8 55 12 21 75 .) Use a GDC or graphing software to create two data plots: speed versus thinking distance and speed versus braking distance. Describe your results. Graph 1 The given values in Table 1 of speed (km/h) and thinking distance (m) were plotted against each other in this graph. With close observation, it is noticeable that the speed is proportional to the thinking distance. In other words: the points construct a straight line that is going through the origin. In this example, we are able to assume that the more the driver increases his/her speed, the more time it takes him/her to apply the brakes. Graph 2 In graph 2, the values of Table 1 of speed (km/h) and braking distance (m) were plotted against each other in this graph. Unfortunately, no straight line can be drawn as the thinking distance is increasing more than proportional. From the 1st point to the 2nd point it is increasing by 8, from the 2nd to the 3rd by 10, from the 3rd to the 4th by 14....and so on. The conclusion of this graph is that the faster the car is moving, the longer it takes the car to brake. 2.) Using your knowledge of functions, develop functions that model the behaviours noted in step 1.

#### Female BMI

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.

#### Random Variables

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

#### Math Investigation - Properties of Quartics

Investigation Mathematical Investigation Ritesh P. Kothari 0/25/2007 A quartic function with a "W" shape has two points of inflection, Q and R. In this investigation a line is drawn through Q and R to meet the quartic function again at P and S. The ratio PQ: QR: RS is to be investigated using specific examples to form a conjecture, and then examined formally to prove the findings. IB Mathematics Portfolio Type I Candidate name: Ritesh Kothari Candidate Code: 001859-018 School name: Mahatma Gandhi International School School Code: 001859 . Graph the function f(x) = 4 - 8 3 + 182 - 12 + 24 The above graph is made using "Microsoft Excel 2007" and the displayed function is F(x) = 4 - 83 + 182 - 12 + 24 Scale used: X: - 1.5 to 5.5 Y: 6 to 66 2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersects the quartic function again, and calculate the ratio PQ:QR:RS. Planning: * Find the points of inflection in the function by differentiating the function twice. * Substituting the X values into the function to get Y co-ordinates. * Get the equation of a line which passes through the inflection points, named Q and R respectively. * Find out two points P and S, where f() intersects the line which passes through the inflection points (Q and R). * Finding out the roots of the function, ignoring the found

#### Properties of Quartics

Table of Contents Introduction In this investigation quartic functions will be explored, in particular quartic functions with a unique "W" shape. These quartic functions will have two points of inflection, which will be referred to as Point Q and Point R. When Line QR is drawn it intercepts the quartic function twice more; those interceptions will be referred to as Point P and Point S. The ratio of PQ:QR:RS will be investigated and any findings will be formally proven. This investigation will include the given function f(x) = x4 - 8x3 + 18x2 - 12x +24 and the chosen function f(x) = x4 + 3x3 -5x . These functions will be depicted with graphs. Method . Graph the function f(x) = x4 - 8x3 + 18x2 - 12x +24. 2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersect the quartic function again, and calculate the ratio PQ:QR:RS. Find Q and R Find First Derivative: f(x) = x4 - 8x3 + 18x2 - 12x +24 f'(x) = 4x3 - 24x2 + 36x - 12 Find Second Derivative f"(x) = 12x2 - 48x + 36 Set to Zero 0 = 12x2 - 48x + 36 Apply Quadratic Equation ; x = 1, 3 Concavity Checks f(0) =36 positive A switch from positive to negative indicates an inflection point. f(2) = -12 negative A switch from negative to positive indicates an inflection point. f(4) = 36 positive Find Y Point Q f(1) = 23 Point R

#### Matrix Binomials

Introduction The goal of this portfolio assessment is to find an expression for Xn, Yn, (X + Y)n, [An, Bn, (A + B)n] in both questions whilst expressing them: Mn in terms of aX and bY. The purpose of this assessment is to find out how we can interpret matrix binomials using different values and similarities to find the pattern occurring. We've been given a general statement to express Mn in terms of aX and bY, to do so we must substitute a into matrix X to get a new matrix 'A', and b into matrix Y to get the new matrix 'B'. The task given now is to see if the pattern really did work with other numbers, and to prove the general statement. * Question 1 Let X = and Y =. Calculate X2, X3, X4; Y2, Y3, Y4. By considering integer powers of X and Y, find expressions for Xn, Yn, (X + Y)n. Alright now to calculate X2, X3, X4; Y2, Y3, Y4, I will firstly show how these matrices are multiplied, and then I shall use my graphics calculator to do the rest. As doing so I will also look for a pattern trend in which I can use to relate to fine the expression Xn, Yn, (X + Y)n. By doing so I will carefully look at how the matrix trend is created, therefore making it easier to find the expression. I found that I can use a specific matrix property in order to find the expression as well as the arithmetic progression. This will ultimately determine how the expression is achieved and if it's

#### IB Math Portfolio: Light of My City

IB Math HL Portfolio Type II: The Light of My City For this project, I chose the glorious city of Pyongyang, the capital city of what is now the Democratic People's Republic of Korea. Other than its historic importance and communist architectural grandeur, Pyongyang is noted for having a clear distinction during its four seasons, and a clear variation of sunlight depending on the season and time. In regards to the data, I used the year of 2008 as the basis for the calculations and data. PART I: The Data Find the shortest day of the year and the amount of sunlight on that day. The shortest day of the year coincides with the December solstice which occurred on Sunday, December 21st in the year of 2008 at 9:04 PM. At this time, the length of the day (the amount of sunlight) totally at exactly 9h 25m 41s from 7:52 AM when the sun rose to 5:18 PM when the sun set. Find the longest day of the year and the amount of sunlight on that day. The longest day of the year always coincides with the June solstice which occurred on Saturday, June 21st in the year of 2008 at 8:59 AM. The length of the day totaled at precisely 14h 54m 33s, from 5:11 AM when the sun rose, to 8:06 PM when the sun set. The difference as we can see between the two extremes of the year is 5 hours 28 minutes and 52 seconds, which is quite a difference. year Equinox Mar Solstice June Equinox Sept

#### Mathematics SL Portfolio Type II Modeling the amount of drug in a bloodstream

Mathematics SL Portfolio Type II Modeling the amount of drug in a bloodstream November 30, 2007 The use of math in life is extremely beneficial; one can know or at least predict what to expect, why to expect it and how to find it. An example of an application of math in life would be medicine; doctors can calculate how much medicine to prescribe and how long it lasts or how long it takes for it to decay, therefore, how often should the patient take it. In a case for treating Malaria, an initial does of 10 milligrams (µg) of a drug was given and an observation for a total period of 10 hours was done, the amount of drug was measured every half an hour, the results were plotted on the following graph(figure 1): As can be seen, this graph shows the rate of the breakdown of the drug in the blood where the amount of drug in the bloodstream decreases with time; they are inversely proportional. In the following table are the numerical results of this observation taken from the previous graph: Time in hours (t/hr) Amount of drugs in µg (Q/µg) 0.5 9.0 .0 8.3 .5 7.8 2.0 7.2 Time in hours (t/hr) Amount of drug in µg (Q/µg) 2.5 6.7 3.0 6.0 3.5 5.3 4.0 5.0 4.5 4.6 5.0 4.4 5.5 4.0 6.0 3.7 6.5 3.0 7.0 2.8 7.5 2.5 8.0 2.5 8.5 2.1 9.0 .9 9.5 .7 0.0 .5 By making the previous observation, more information can be concluded, however, in

#### Math IB SL BMI Portfolio

This portfolio is an investigation into how the median Body Mass Index of a girl will change as she ages. Body Mass Index (BMI) is a comparison between a person's height (in meters) and weight (in kilograms) in order to determine whether one is overweight or underweight based on their height. The goal of this portfolio is to prove of disprove how BMI as a function of Age (years) for girls living in the USA in 2000 can be modeled using one or more mathematical equations. This data can be used for parents wanting to predict the change in BMI for their daughters or to compare their daughter's BMI with the median BMI in the USA. The equation used to measure BMI is: The chart below shows the median BMI for girls of different ages in the United States in the year 2000: Age (years) Median 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 values "height (m)" and "weight (kg)" can be discarded when trying to analyze this data because it is not consistent that a girl will have a fixed weight to height or vice versa when they are a certain age. For example, the median BMI for a 10 year old girl is 16.80. With only the data provided, one cannot isolate either the height variable or the weight variable using the formula for calculating BMI. Using

#### Infinite surds portfolio - As you can see in the first 10 terms of the infinite surd, they are all irrational numbers.

MATH PORTFOLIO INFINTE SURDS Submitted By Tim Kwok Math 20 IB Presented To Ms. Garrett April 27, 2009 TABLE OF CONTENTS Introduction to Surds and Infinite Surds Page 2 Infinite Surd Example 1 Page 2 * First Ten Terms of Sequence Page 2-3 * Formula for the Following Term Page 3 * Graph of First Ten Terms Page 4 * Relation Between Terms and Values in Infinite Surd Page 4 * Exact Value of the Infinite Surd Page 5 Infinite Surd Example 2 Page 6 * First Ten Terms of Sequence Page 6-7 * Formula for the Following Term Page 7 * Graph of First Ten Terms Page 8 * Relation Between Terms and Values in Infinite Surd Page 8-9 * Exact Value of the Infinite Surd Page 9 Infinite Surd Example 3 Page 10 * General Form of Infinite Surd Exact Value Page 10 Infinite Surd Example 4 Page 11 * Values That Make an Infinite Surd an Integer Page 11 * General Statement for Values That Make an Page 12 Infinite Surd an Integer * Limitations to the General Statement Page 13 References Page 14 Surds are used commonly in math, they just are not referred to as surds. A surd is any positive number that is in square root form. Once you simplify the surd it must form a positive irrational number. If a rational number is formed, it is not considered to be a surd. Infinite surds are just surds forming a sequence that goes on forever. The exact value of an infinite surd is