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International Baccalaureate: Maths
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Moss's Egg. Task -1- Find the area of the shaded region inside the two circles shown below. The two large circles have a radius of 6cm.
The area of the small circle can therefore be calculated using the formula indicated: , where A equals the area and r is the radius. Thus: A = (32) = 9 � 28.3 cm2 Task -2- The same circles are shown below. Find the area and perimeter of the triangle ABC. a) In order to determine the area of triangle ABC, we must adopt the formula: , where b is the base and h is the perpendicular height. From the information we are given, we know that the base of the triangle is 6 cm in length, as we know that this is the diameter of the small circle and that the base extends to its ends (A to B).
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Linear Function: y=ax+b First, I use two points of (1955, 609) and (1985, 1070) from the given data to find the parameters of the function. I chose these two points because they are the only two combinations that are all integers, which will be easier to calculate. 609=1955a+b 1070=1985a+b Finding value a and b by using simultaneous equation: 1070=1985a+b - 609=1955a+b ________________________________ 461= 30a a = ?15.37 b = 609-1955�15.37=29439.35 Which gives the equation of y=15.37x-29439.35 as below: Graph 2: Population of China 1950~1995 with linear function It appears to be rather fit to the coordinates.
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It is of course not perfectly linear because the growth of a population would not be perfectly linear. This data is close to linear but it is not perfectly linear. A linear function is plausible. This is a graph with the data points on it as well as a linear function: y = 15.496x - 29690.2501 The function of the parent linear function is If in the TI-84 calculator you do a linear regression. You first need to put all the data from the years 1950 to 1995 into L1 and the corresponding population data into L2. After you have put the data in the L1 and L2 lists. After the data has been entered into the lists.
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On the middle part an equilateral triangle is drawn. And this continues in the following stages. Below are the values for the first 4 diagrams, . n Nn Ln Pn An 0 3 1 3 1 12 4 0.57735 2 48 0.64150 3 192 0.67001 (5 s.f) Number of sides Initially there is an equilateral triangle at stage n=0, each of those sides is divided into 3 sides. And there is another equilateral triangle created at the center points. Thus from that I can conclude that each side becomes from sides. as the diagrams are given to us, for the first 4 stages I counted the sides.
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tn), it is clear that a quadratic equation is formed by this relation, because the second differences are constant and not zero. n tn First differences Second differences Third differences 1 1 / / / 2 3 2 / / 3 6 3 1 / 4 10 4 1 0 5 15 5 1 0 6 21 6 1 0 7 28 7 1 0 8 36 8 1 0 Let f equal the value of tn . Consider any quadratic equation which can be represented as.
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Stellar Numbers Portfolio. The simplest example of these is square numbers, but over the course of this investigation, both triangular numbers and stellar numbers will be looked at in greater depth.
Complete the triangular numbers sequence with three more terms. The given diagrams for the problem were through and are shown below. The next three terms needed to complete the diagram are through . These are illustrated in the pictures below. The white lines on bottom of each diagram are the dots that need to be added to go from one triangular number to the next. Question 2 Find a general statement that represents the nth triangular number in terms of n When completing different trials in attempts to find the general term for Triangular Numbers, the following information was determined.
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Logarithms. In this investigation, the use of the properties of logarithms will be used to identify patterns, relationships, and limits of logarithms and sequences.
A geometric sequence is a sequence whose consecutive terms are multiplied by a fixed, non-zero real number called a common ratio. Therefore, to prove that the sequence is indeed, geometric, the common ratio must be found. When the sequence is examined closely, you will notice that the log number remains the same throughout the sequence and that it is the base of the logarithm that changes with every term. Understanding this, you can use the bases to find the common ratio of the sequence by dividing the consecutive terms by the terms preceding them.
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Stellar Numbers Portfolio. In this task I will consider geometric shapes, which lead to special numbers
The complete pattern in this 6-stellar is at follows: 0+1= 1 1+12=13 13+24=37 37+36=73 Therefore, S5 = 73+ 48=121 S6 = 121 + 60= 181 Similarly to the triangular numbers, the 6-stellar sequences uses the same method but this time the number of dots can be found by preceding term added to by the multiples of 12 (as shown in red). Other way to write the pattern is in this way: (1+0(12)), (1+1(12)), (1+3(12)) , (1+6(12)) , ... As soon as I wrote it this way I realized that there is clear relationship between the Triangular numbers and the way the 6-stellar numbers term.
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Systems of Linear Equations. Investigate Systems of linear equations where the system constants have well known mathematical patterns.
As we can see in the graph the two equations intersect at a point and that point is the same as the solution we found when we solved the problem algebraically. So the point that the two equations intersect is (-1, 2). Graph Here are some examples using the same pattern: Example 1: x + 3y= 5 x= 5 - 3y x= 5 - 3(2) x= -1 5x - y= -7 5(5 - 3y) - y= -7 25 - 16y= -7 -16y= -32 y=2 y= 2 Example 2: 2x + 6y= 10 x= 5 - 3y x= 5 - 3(2 )
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Stopping distances portfolio. In this task, we may develop individual functions that model the relationship between speed and thinking distance, as well as speed and braking distance. We could also develop a model for the relationship between speed and ov
Initial Data Set The table below lists average times for these processes at various speeds. Speed (km/h) Thinking distance (m) Braking distance (m) 32 6 6 48 9 14 64 12 24 80 15 38 96 18 55 112 21 75 Speed vs. Thinking distance As it is visible from the table below when speed doubles, thinking speed doubles, and when speed triples, thinking speed triples. There is a clear linear relationship which can be found from any 2 points, like (64, 12) and (96, 18). Then if we consider m as a stable ratio, so the following relation is found to be true: then and = 0.1875 Speed (km/h)
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Infinite summation portfolio. A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation
However, the numbers become so small, that they become insignificant, or in other words they are equal to 0. Now, we need to find the sum of Sn : Now, using Excel 2010, let's plot the relation between Sn and n : Looking at the graph, we can notice that Sn increases rapidly at first, and then it evens out when it reaches 2, which seems like an asymptote. The same happens with the terms' values. They decrease rapidly until they reach the 0, which if we plot will seem like its asymptote. Therefore, we can see that both move a maximum of 1 unit away from their first point, and then even out to the mentioned asymptote.
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Infinite summation SL Portfolio type I. Concerning the portfolio, the evaluation is composed on the sum of the series
Microsoft Excel Relation between and ( 0 1 2 1.000000 1 1 2 1.693147 2 1 2 1.933374 3 1 2 1.988878 4 1 2 1.998496 5 1 2 1.999829 6 1 2 1.999983 7 1 2 1.999999 8 1 2 2.000000 9 1 2 2.000000 10 1 2 2.000000 I can notice right now from this plot, that the value is increasing when the value increases, so simply they are positively correlated. however, the graph does not go beyond .
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This project was given to us, as we are currently working on and will soon be completing a unit based on the calculation of surface area and volume for 3-d and 2-d shapes. Conclusively, all those that have attempted the completion of this project, will have an end result of exactly which is the 'best' container for mass production. 4 Procedures, materials and methods Materials For the completion of this project there are some necessary materials that every student or pupil wanting to complete this project must have: - GDC (Graph Display Calculator) - "Graph" (www.padowan.dk) or "Grapher" (Apple Applications)
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AIMS * To know if the number of accidents encountered by vehicles on the Accra Kumasi highway are related to number of times the drivers maintained their vehicles, the average speed at which they drove their vehicles and their ages. * To find out the strength of the relationship that exist between the number of accidents encountered by the commercial vehicles, and the number of times they maintained their vehicles, the average speed at which they drove their vehicles and their ages if there exists any at all.
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Mathematics SL Portfolio Type II. This portfolio considers commercial fishing in a country in two different environments, namely the sea and fish farms. The statistics are obtained from UN Statistics Division Common Database from the year 1980 to 2006. T
Part I Defining Variables, Parameters and Constraints There are two variables in the data given. These two variables affect each other's value. The independent variable is the year and the dependent value is the total mass of fish caught in the sea shown in thousands of tonnes. The change in year affect the total mass of fish caught. The year increases by one. Parameters in the context of mathematical model are constants involved in the relation between the independent and dependent variable (Bard, 1974). Possible parameters that may affect the fish production are weather, government policy, technological advancement, environmental factors and demands.
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Infinite Surds Investigation. This graph illustrates the same relationship as was demonstrated in the infinite surd of 1. The largest jump between values occurs between 1 and 2.
9 10 -0.000012055 10 11 -0.000003752 The exact value of this infinite surd could be found by using the idea that: Now substitute the value for an+1 into the equation and solve for an by setting it equal to zero. The surd is canceled once the whole equation is squared and this is left: Now this is solved using the quadratic formula: The negative answer, , must be disregarded because a negative answer for a surd is not possible. Therefore the exact value of the infinite surd is: For the infinite surd expression: This would be the sequence of terms for an: etc.
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Fishing Rods Portfolio. Leo has a fishing rod with an overall length of 230 cm. The diagram shows a fishing rod with eight guides, plus a guide at the tip of the rod.
The graph of the data would look like this: Quadratic Function: (ax2+bx+c) To find a quadratic function for this data, I used the first three guide numbers and distances from the tip: (1, 10), (2, 23), (3, 38) and used these numbers in the matrix method To find [X], we could use this formula: Quadratic function is: Cubic Function: ax3+bx2+cx+d The same method was used to find the cubic formula, except that the first 4 guide numbers and distances were used in the matrix method.
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To try and detect a correlation, this can be attempted with other values of n, using the same limits. For n=3, between x=0 and x=1: For n=4, between x=0 and x=1: Further examples: Thus far, we have a correlation that for power n, the ratio of areas is also equal to n. Before making a conjecture, we should examine whether this occurs with other limits as well. For n=2, between x=0 and x=2 (note that the limits on y change for the calculation of A, but the process remains otherwise the same): For n=3, between x=0 and x=2: For n=4, between x=0 and x=2: I then tested the results for when.
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Modelling Probabilities in Tennis. In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity.
Let X denote the number of points scored by A. We can now state that: and therefore that: . We can hence easily calculate the probability distribution for all possibly values of X: x P(X=x) 0 0.000017 1 0.000339 2 0.003048 3 0.016258 4 0.056902 5 0.136565 6 0.227608 7 0.260123 8 0.195092 9 0.086708 10 0.017342 This can be presented as a histogram: This shows that the modal score will be 7, with the highest individual probability. Based on the binomial distribution, we can also calculate the expected value and standard distribution: Based on this we can see that most scores fall between 4 and 8, with the mean score being 6.6667.
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Stellar numbers and triangular numbers. Find an expression for the 6-stellar number at stage s7. Find a general statement for the 6-stellar number at stage sn in terms of n.
are numbers that can be displayed in the form of a triangular pattern composed of evenly spaced dots. This can be seen below. TASK 1: Complete the triangular number sequence with three more terms. Find a general statement that represents the nth triangular number in terms of n. Given the fact that our current sequence is 1, 3, 6, 10, 15... we need to find the nth term. Looking at the sequence, we see that there difference between each term increases by 1 so for example the difference between u1 and u2 is 2 and the difference between u2 and u3 is 3.
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In this task, we are required to investigate the mathematical patterns within systems of linear equations. We need to concept of matrices and algebraic equations in this task.
To check the solution The solution of this 2�2 system of linear equations is unique. 1. Substituting into equation (1): The solution is x = -1, y = 2. 2. ( Substituting into equation (1): The solution is x = -1, y = 2. 3. Substituting into equation (2): The solution is x = -1, y = 2. 4. (2) �: (3) - (1): Substituting into equation (1): The solution is x = -1, y = 2. 5. Substituting into equation (1): The solution is x = -1, y = 2. From the five 2�2 system of linear equations I have investigated, all of them have a unique solution of x = -1, y = 2.
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The year will be represented by x and the population will be represented by y There are restrictions that also need to be set; the year as well as the population can never be anything below 0. My parameter for time will be that for each year, "t" will equal the number of years after 1950. Therefore, for 1950, "t" will equal 0, for 1955 "t" will equal 5 and so on. Below is a graph that plots the above data points.
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Math portfolio stellar numbers. This assessment will investigate geometric shapes that lead to special numbers.
Expressing this mathematically, Rn = Since Rn=2Tn one can substitute Rn in order to get Tn, 2Tn= 2Tn=n(n+1) Tn= Thus it seems like a general statement that represents the nth triangular number in terms of n. After investigating the triangular shapes, more complex Stellar shapes will be investigated. And so forth... Each star has a number of vertices these will be labelled p. Every value of p leads to P-Stellar number, labelled Sn. In this case it is a 6-stellar number.
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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.
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 transversal. Number of Horizontal Lines Number of transversals Number of Parallelograms formed First difference between terms Second difference between terms 2 2 1 2 3 3 2 2 4 6 3 1 2 5 10 4 1 2 6 15 5 1 2
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fishing rods portfolio (SL maths)This portfolio deals with Leo's fishing rod which has an overall length of 230 cm together with eight guides that are placed a certain distance from the tip
and (Distance from tip in centimeters). The table below shows these variables. variable - Guide Number from the tip 1 2 3 4 5 6 7 8 variable - Distance from the tip (cm) 10 23 38 55 74 96 120 149 The constraints or parameters for the two variables are: * variable parameter = 1 ? ? 8 * variable parameter = 10 ? ? 149 Furthermore, the variable, is the Independent Variable, and the variable is the Dependant Variable. Table 1, which gives the distances for each of the line guides from the tip of Leo's fishing rod, is plotted on the graph below: I will consider the variables as a sequence, and from there I shall calculate the polynomial equation modeling this situation.
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