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International Baccalaureate: Maths
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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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Parallels and Parallelograms. Aim: To find the effects of increasing the number of intersecting transversal and horizontal parallel lines on the number of parallelograms formed.
C U D, D U E, A U B U C, B U C U D, C U D U E, A U B U C U D, B U C U D U E, A U B U C U D U E. 7 lines A B C D E F This would give us 21 parallelograms: A, B, C, D, E, F, A U B, B U C, C U D, D U E, E U F, A U B U C, B U C U D, C U D U E, D U E U F, A
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Stellar Numbers. Aim: To deduce the relationship found between the stellar numbers and the number of vertices in the stellar shapes that dictate their value.
Let us look at the table again: Rows Dots 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 In this table, however, instead of referring to simply 'term' we consider the number of rows, since that is a varying pattern that logically coincides with the concept of 'term'. From this we can get the general expression: 1. At first, the number of rows and dots do not seem to exhibit any discernible relation.
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Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes with p vertices leading to p-stellar numbers were to be considered.
= 10 T5 5 + (1 + 2 + 3 + 4) = 15 T6 6 + (1 + 2 + 3 + 4 + 5) = 21 T7 7 + (1 + 2 + 3 + 4 + 5 + 6) = 28 T8 8 + (1 + 2 + 3 + 4 + 5 + 6 + 7) = 36 As seen in the diagram above, the second difference is the same between the terms, and the sequence is therefore quadratic. This means that the equation Tn = an2 + bn + c will be used when representing the data in a general formula.
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Parallelograms. This investigation will focus on the number of parallelograms formed by intersecting lines also knows as transversals.
A4 Set Notation: A1, A2, A3, A4, A5, A1 ? A2, A2 ? A3, A3 ? A4, A4 ? A5, A1 ? A2 ? A3, A2 ? A3 ? A4, A3 ? A4 ? A5, A1 ? A2 ? A3 ? A4, A2 ? A3 ? A4 ? A5 and A1 ? A2 ? A3 ? A4 ? A5 Set Notation: A1, A2, A3, A4, A5, A6, A1 ? A2, A2 ? A3, A3 ? A4, A4 ? A5, A5 ? A6, A1 ? A2 ? A3, A2 ? A3 ? A4, A3 ? A4 ? A5, A4 ?
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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
I know that the length of the cuboid is 150 meters. So I have to find out the height and the width of the cuboid. The diagram below shows a cuboid which is fitted inside the curve roof structure. Let ABCD be the largest possible cuboid which can be fitted inside this curved roof structure I will let "2V" be the width of the cuboid and "H" be the height. The parabola is symmetrical structure, so do the cuboid, so I have taken the width as "2V" that is the width "V" is on the left side and "V" is on the right side of the axis of symmetry of the parabola.
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However, after that, an-an+1 value has been had no huge change which means that difference is close to 0. Apply Here is a proved formula a= a2=1+a a2-a-1=0 Use quadratic equation a= = 1.618033989 or -0.6180339887 However, the root cannot be the negative number So, ? 1.618033989 Consider another infinite surd where the first term is Repeat the entire process above to find the exact value for this surd.
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Function that best models the population of China. Some of the functions that I think that could model this data were linear function () and exponential function ().
As we can see that the population is increasing as by the time. As the years are passing by, the trend of the population is gradually increasing more and more, the graph is becoming steeper. There is no rapid growth or decline in the population. I think that this is in between gradual and rapid growth of the population. Just after the year 1965, there is a tiny bounce that can be seen that pushes the growth rate more. To help see, I have added here two lines that tells us more about the slope.
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After seeing that my result matches the results in Excel, I decided to then find out the sum. I found out the sum using my TI-84 Plus. The method is shown in the appendix. This is what I did in order to get for . <-- I got 1 because of . <-- Here is equal to plus because is the sum of the pervious term. So if I add which is term number 1, it will give me the . I will use this formula to find out the sum's up to n = 10. <-- Now, we can sub as 1 and as Here n = 2, so I did this for where On the other hand, I used Microsoft Excel 2010 to do it for me too.
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Examples 1. Find the y-intercept for the following equation. * 2. Find the y-intercept for the following straight line. Slope/Gradient The slope (gradient) ultimately determines the 'steepness' or incline of a line, the higher the slope, the steeper the incline will be. For example, a horizontal line has a slope equal to zero while a line with an angle of 45o has a slope equal to one. The sign (positive or negative) of the slope is very important, as it determines whether the line slopes uphill or downhill.
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+ 6 = 15 + 6 = 21 7th number: (6th triangular number) + 7 = 21 + 7 = 28 Let the triangular number be. n 1 2 3 4 5 6 7 1 3 6 10 15 21 28 ??????? This method of calculating the values of can be proven to be accurate using the method of differences. Here, the difference between consecutive values of are calculated, and the data collected from that is manipulated in a manner to prove that the earlier used method of calculating subsequent values of is reliable.
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In this essay, I am going to investigate the maximum number of pieces obtained when n-dimensional object is cut, and then prove it is true.
To begin the solution, consider the results for n=1, 2,3,4,5. Let R represent the maximum number of pieces in n-cuts of a line segment. The value for R is shown in the table. n 1 2 3 4 5 R 2 4 7 11 16 1Recursive rule: R1=2 R2=4=2+2 R3=7=4+3 R4=11=7+4 R5=16=11+5 ... Rn=Rn-1+n[U1] Assume that R0=1, then Rn=1+1+2+3+......+n Rn=1+(1+2+3+......+n) =1+2+3+......+n=n/2[2a+(n-1)*[U2]d][U3], a=1, d=1 1+2+3+......+n=n/2[2+(n-1)]=n/2[n+1]=(n^[U4]2+n)/2 Therefore, the recursive rule to generate the maximum number of regions is 1+ (n^2+n)/2= (n^2+n+2)/2[U5]. When n=5, R5=(5^2+5+2)/2=16, which corresponds to the tabulated value for n=5 above.
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For example, the value for equals to 15 +6 which is 21. The same applies for the other values of . The next three terms are represented by this diagram: 6 7 8 Shape Number of dots 21 28 36 When we want to find the expression for this sequence, we have to know which difference is constant. Seq.: 1, 3, 6, 10, 15... 1st diff.: 2 3 4 5... 2nd diff.: 1 1 1... We found from the above that the second difference is constant. Therefore the expression will be .
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This is important because if this sequence diverges, the general statement would be . A convergent series is a series in which the terms decrease in magnitude rapidly and for which the sum of the first several terms is not too different from the sum of all of the terms of the series. The following is an example of a divergent series. = Thus, there is no general statement to represent the infinite sum of this sequence. In order to detect whether the given sequence is a convergent sequence or not, there are 3 tests that can prove it.
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This is due to the fact that the amount of time humans can tolerate is dependent of the G-force itself; therefore . The notation in which the function would model after is. We set that because this function would only be applied to positive G-Force. Another constraint is that because the human tolerance time has to be at a value which we can measure. When table of values are plotted on a graph, it is clear that as the horizontal G-forces increase, the time would quickly decrease as a consequence.
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other > The overall length of the fishing rod - it cannot be negative or too long as this will reduce efficiency The data points given above can then be plotted onto a graph using Microsoft excel. Using Matrix methods we can find a quadratic function to model the situation from the given data points: A quadratic equation is in the form. As there are 3 unknown variables (a, b, c) we can create a 3 x 3 matrix and a 3 x 1 matrix to model the given information.
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Infinite Summation Internal Assessment The idea of this internal assessment is to investigate the effect changing the value of x and a have on the graph of the general sequence given.
The next sequence will be a similar concept of the previous one, but the change in values of x=1 and a=3 which is 1, , , ... n Sn 0 1.000000 1 2.098612 2 2.702087 3 2.923082 4 2.983779 5 2.997115 6 2.999557 7 2.999940 8 2.999993 9 2.999999 10 3.000000 = This table show the value of , which is the sum of the sequence and never exceeds 3. This may be similar to the previous sequence but it varies as the increase faster throughout the same value of as reaches to 3.
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Normally when you encrypt something, you would most commonly have "units" substituting your letter or sentences. The units could be in one's, two's or even three's. There are various ways of substitution cyphering, one way (the most common way), would be to use ROT13, an alphabet rotated over 13 steps. Simple Hill Cypher, is a method to solve and encrypt a message. The method consists of using substitution. Eg: A B C D E F G H I J K L M N O N O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 14 15 P Q R S T U V W X Y Z .
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A graph of the displayed functions is below this (Figure 2.2). An explanation for each significant situation is provided below Figure 2.2. Figure 2.1 Situation # 2 1 3 4 4 2 5 6 7 8 9 10 [tt2] Situation 3: This situation caused me to hypothesize the conjecture that, where A is the A value from the equation. In addition, g(x) was tangent to f(x) at the point (10,10). This meant that x2 and x3 were the same (10), and showed that the conjecture held true for tangents. Situation 6: A concave down parabola in the 1st quadrant still holds the conjecture, provided that the conjecture is changed to Situation 7: Irrational A values work for the conjecture.
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