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International Baccalaureate: Maths
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LACSAP FRACTIONS - I will begin my investigation by continuing the pattern and finding the numerator of the sixth row.
Sn = ( 2u1 + (n – 1)d) Sn = ( 2(1) + (n – 1)1) Sn = (2 + n – 1) Sn = ( n + 1) ï General Statment = 0.5n2 + 0.5n Next, to make sure my general statement is valid I will test it’s validity and figure out the numerator value for the sixth and seventh row. Test validity for when n is equal to 5 Numerator = 0.5n2 + 0.5n = 0.5(5)2 + 0.5(5) = 0.5(25) + 2.5 = 12.5 + 2.5 Numerator = 15 when n is 5 the numerator is 15 Find numerator(N)
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In this portfolio, I am required to investigate the number of regions obtained by making cuts in one, two and three dimensional objects
After noting this simple pattern, I derived the recursive rule as follows ? No of segments = No of cuts made + 1 Two ? dimensional object ? For a circle, it was hard to find the maximum number of regions because chords can cut different number of regions in the way they are drawn and because I needed maximum number of regions, I had to draw chords in a specific manner. The diagrams are as follows ? No of chords (n)
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This will form an equation 2y=3+1 When -1 one is taken on the other side and the final equation would be 2y=4 y= y=2 Hence the point of intersection of the linear equations is (-1, 2) In the similar way if we consider another pair of linear equations where the constants are forming A.P. -x-34y=-68 2x+17y=32 In this equation we have to double the second equation to eliminate the y term and add x terms and the answers from which we will get -x-34y=-67 4x+34y=64 = -x+4x=64-67 3x=-3 x= x=-1 Now using the value of x we can find the value of y by replacing the x term with -1 2(-1)
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To prove this relationship between the two, we have to first determine the corresponding sides and angles in the triangle doing as such: For the sides we have: AO is corresponding to OP?, where AO and OP? are the bases, OP is corresponding to AP? being the right sides, and for the left we have AP corresponding to AO. Meanwhile, the angles are as such: angle OPA corresponds to angle P?AO, angle P?AO corresponds to angle AOP?, and AOP corresponds with OP?A.
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To achieve these numbers plug in the data in the form of linear regression, where the height (y-value) is the dependent variable while the year (x-value) an explanatory value or independent. Also, the value (coefficient of determination) is used in statistics whose main purpose is to predict the future outcomes on the basis of related information. In the case of linear regression, between the outcomes and the values of the single regressor being used for prediction. In such cases, the coefficient of determination ranges from 0 to 1. Contextually, the closer the value is to the value of 1, the more confidence we have in (accurate)
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Both values have to be >0. The behaviour of this graph can be modelled by the sine function , or by the cosine function as it is periodic and ondulating, which means it repeats a pattern as it goes up and down. Other functions can?t be used in this model as the information given can be reflected in other type of graphs. Finally, we can deduce that it will be a cosine function as the portion that we observe in tha graph doesn?t go through the origin (0,0), as the sine function would do.
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DIAGRAM Now I am going to find the dimensions of the cuboid with maximum volume which would fit inside this roof structure. The length of the cuboid is already been provided to us which is150 meters now we only have to find its height and breadth. Draw a cuboid within the parabola structure ABCD is the largest cuboid possible that can be fitted in the curved structure. Let the breadth of the cuboid be ?2x? and the height be ?h We know that the width of the structure is 72 meters.
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and adding 1, as shown here: 15=2×10-6+1 A second example, the calculation of the numerator of the third row: k3=2×6-3+1=10 A third example, the calculation of the numerator of the fourth row: k4=2×10-6+1=15 And the general formula for the numerator of the nth row is: kn=2×kn-1-kn-2+1 2.) Considering Table 2 it can be concluded that the numerator of each row can be calculated by multiplying two arithmetic sequences where one factor of the formula is a multiple of 0.5 and the other factor can be calculated by adding 1 to the row number of the numerator we are looking for.
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IB Math SL Portfolio The aim of this task is to investigate positions of points in intersecting circles.
Hence the sum of the angles of A and O has to be greater than P. Angle Measurement: A= 75.522 degrees O= 75.522 degrees P= 28.956 degrees Radius equals 1, OP is equal to 3, so the other side length must be equal to 3 because this is an isosceles triangle. Use Law of Cosines To Find Angle: Cos O = ------ Where A= 3, O = 3, P = 1. Where A=A, B=O, C=P Cosine O= O= O = 83.621 Degrees Use Law of Sines to Find Other Angles: SO Sin A= .9994 A= A= 83.621 degrees Sum of
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from x=a to x=b using ?n? number of trapeziums. Investigation: To find the area under the curve f(x) =+3 with domain [0,1], I must first plot the function. When graphed, the function looks like this: Case1: n=2 To find the approximate area that exists under the curve, I will first divide it into two trapeziums by graphing x=0.5. This will divide the curve into two equal parts. Using technology, I will join the 4 coordinates of each half of the curve, thereby obtaining two trapeziums. Calculating and adding the area of the two trapeziums will give me the approximate area of the space present beneath the curve.
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The sequences are: 2 (+2) 4 (+3) 7 (+4) 11 4 (+2) 6 (+3) 9 7 (+2) 9 Figure 4 Denominator pattern As we can see in Figure 3, the denominator increases by 1 starting with a step of 2. Following this rule, the denominators are: 6th row: 16 - 13 - 12 - 13 - 16 7th row: 22 ? 18 ? 16 ? 16 ? 18 ? 22 Since I have already found out the numerators (Step 1), the complete 6th and 7th row are as followed: 6th row : 1 1 7th row : 1 1 Step 3: Step 3 is about finding a general expression for the numerator.
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The numerator of the sixth row is as follows: 1 21 21 21 21 21 1 1. Using technology, plot the relation between row number, n, and the numerator of each row. Describe what you notice from your plot and write a general statement to represent this. The technology used to find any general statement is by using Graphic Display Calculator, TI-84. The programme used is Quadratic Regression which is a programme that finds the equation of the parabola given a set of data. I denote y-axis as N (numerator)
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The purpose of this investigation is to create and model a dice-based casino game using probability. In order to be successful, this game must be able to allow the casino to profit from running it,
Therefore, the probability that player A wins is the probability that the two players roll different numbers and that player A?s number is the higher of the two. As each die has faces numbered from one to six and there are two players present in the game, there are possible outcomes for the game. Of these 36 outcomes, there are six ways for players A and B to roll the same number (they can both roll a 1, 2, 3, 4, 5, or 6).
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The t th term will be used to indicate the used term and Sn states the sum of the term given by n. (all in 6 decimal places, whereas the space after the 6 decimal indicates the minor differences between the last 4 sums). S0= t0=1 S1= 1+t1= 1.693147 S2=S1+t2=1.933374 S3= S2+t3=1.988878 S4= S3+t4=1.998496 S5= S4+t5=1.999829 S6= S5+t6=1.999983 S7= S6+t7=1.999998 569 S8= S7+t8=1.999999 891 S9= S8+t9=1.999999 992 S10= S9+t10=1.999999 999 5 n Sn 0 1 1 1.693147 2 1.933374 3 1.988878 4 1.998496 5 1.999829 6 1.999983 7 1.999998 8 1.999999 9 1.999999 10 1.999999 This diagram shows the relation between Sn and n using the gained results.
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LAcsap fractions - it is clear that in order to obtain a general statement for the pattern, two different statements will be needed
To start the initial pattern, the pattern is split into two different patterns; one demonstrating the numerators and another denominators. Step 2: This pattern demonstrates the pattern of the numerators. It is clear that all of the numerators in the nth row are equal. For example all numerators in row 3 are 6. 1 1 3 3 3 6 6 6 6 10 10 10 10 10 15 15 15 15 15 15 Row number (n) 1 2 3 4 5 Numerator (N)
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Parabola investigation. The property that was investigated was the relationship between the parabola and two lines that intersected the parabola.
Finally the value of D is found by finding the modulus of Sl - SR. The equation y = x2 ? 6x + 11 is in the form y = ax2 + bx + c. Now the aim is to change the values of a and find a conjecture that can predict the value of D when a varies. HOW WAS THE CONJECTURE FOUND OUT ? If we draw the graph of y = ax2 + bx + c, we see that there are three conditions present in the graph.
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The distance from the tip will be measured in centimeters; the distance is therefore a continuous variable since it is uncountable. One constraint of the data presented in Table 1 is that there cannot be a negative number of guides or a negative distance from the tip; the fishing rod is required to have at least one guide and the distance between two guides has to be a positive number. As a result the plotted graph must be contained to the first quadrant, as the second, third and fourth quadrants contain an axis comprised of negative numbers.
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The 7th row can assumed to be 28 and the 8th row can be assumed to be 36. Let us now unfold the mystery behind this pattern by first finding the common difference. Row Numerator 1st Difference 2nd Difference 1 1 2 2 3 1 3 3 6 1 4 4 10 1 5 5 15 The 1st difference between the numerators does not give us a consistent number thus; there is no consistent increase. However, through the 2nd difference, we get the common difference of 1. Therefore, we can plug it into a quadratic formula; y= ax2+bx+c where y = numerator and x = row number.
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Adding a fifth transversal gives us a total of ten parallelograms. Adding a sixth transversal Figure 5: six transversals A1 , A2 , A3, A4, A5 , A1 á´ A2, A1 á´ A3, A1 á´ A4 , A1 á´ A5 , A2 á´ A3, A2 á´ A4, A2 á´ A5 , A3 á´ A4, A3 á´ A5, A4 á´ A5. Adding a sixth transversal gives us a total of fifteen parallelograms. Adding a seventh transversal Figure 6: seven transversals A1 , A2 , A3, A4, A5 , A6, A1 á´ A2, A1 á´ A3, A1 á´ A4 , A1 á´ A5, A1 á´ A6 , A2 á´ A3, A2 á´ A4,
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Stellar numbers. This internal assessment has been written to embrace one of the rather inquisitive aspects of mathematics, stellar numbers
Now that the reader is familiarized with what triangular numbers are and what they look like, let?s look at the possible patterns which may arise from this sequence of numbers. Table 1 n y 1st difference 2nd difference 1 1 - - 2 3 2 - 3 6 3 1 4 10 4 1 5 15 5 1 6 21 6 1 7 28 7 1 8 36 8 1 Table 1, to the left, shows the results of the data obtained by counting and calculating the number of dots in each triangular number tested, wheren is the term, y
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Lacsap's Fractions Investigation. The way Lacsaps fractions are presented bears a strong resemblance to Pascals Triangle
The difference between each subsequent numerator increases by 1 during each turn, starting from 2. Thus, the numerator of the 6th row will be 15 + 6, which equals 21, as stated above. This proves that there is a significant connection between Pascal?s triangle and Lacsap?s fractions. Figure 3: Graph that plots the relation between row number and numerator The graph above shows the relationship between the row numbers (1-5 on the x-axis) and the numerators (ranging from 1-15 on the y-axis).
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The different variables in the given data are as follows: 1. Independent: The independent variable would be the year. This variable is changed by increasing the year by one, and taking note of the total mass of fish caught in that particular month. 2. Dependent: The dependent variable for this set of data would be the amount of fish caught in each year. This variable is observed by noticing the mass of fish caught after every incremental change to the year. The mass of fish caught can only change when the year has changed, which is why it is dependent on the change of years The parameters or constraints in a particular set of data points are the factors in the mathematical model that are constants related to the independent and dependent variables.
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________________ ________________ Let f(x) = y. ________________ Notice that: ________________ f(2) ? f(1) = 15 ________________ f(3) ? f(2) = 17 ________________ f(4) ? f(3) = 19 ________________ And the difference between these are all 2. ________________ Since after 2 steps of calculating the differences a repeating answer is reached, f(x) is probably quadratic. ________________ See: http://www.purplemath.com/modules/nextnumb.htm ________________ ________________ ________________ Using matrix methods or otherwise, find a quadratic function which models this situation. Explain the process you used. On a new set of axes draw these model functions and the original data points.
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by Gherardo of Cremona, a European mathematician in 1150. Fibonacci (1202) and Robert Recorde (in his work The Pathway to Knowledge published in 1551) all used the English term “surd” to refer to unresolved irrational roots. The radical or root symbol depicts surds, with the upper line above the expression called the vinculum. When expressed using indices, a square root corresponds to a ½ while a cube root corresponds to a 1/3. The following expression is an example of an infinite surd 1+1+1+1+1+… Consider this surd as a sequence of terms an where a1=1+1 a2=1+1+1 a3=1+1+1+1 etc.
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In this task, you will investigate different functions that best model the population of China from 1950 to 1995.
If the population is larger than K, it will decrease. Every positive solution has limt?+? x (t) =K. The logistic model can be normalised by rescaling the units of population and time. Define y := x/K and s := rt. The result is dyds = y (1?y). It is easy to find the explicit solution of the logistic equation, since it is a first-order separable differential equation. The growing solutions are all time-translated versions of the logistic function ys= eses +1=11+e-s which looks like this: This function goes from 0 to 1 as t goes from ?? to +?.
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