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International Baccalaureate: Maths
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This assignments purpose is to investigate how translation and enlargement of data affects statistical parametersI
To find the mean and the standard deviation if subtracting 12 cm from each height, I would first have to subtract 12 cm from each height which is shown in the table below: Then I would repeat the steps I used on the first question using my TI-84 (STAT: CALC: 1-Var Stats: ENTER: List: 2ND: 2: ENTER: Calculate: ENTER), which gives me the following data The mean and standard deviation of subtracting 12 cm in each height gives you 140.9166667 as the mean and 17.08731661 as the standard deviation.
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We take the positive value of the root of the equation, as we are interested in the upper intersection of the two circles, which corresponds to the circle C1. C2=C1 ; The coordinate of midpoint of circle C3 is From the midpoint of triangle C3, we can derive the formula for circle C3 ; OP? intercepts the x-axis when y=0 X=? and Y=0 The magnitude of a vector cannot be 0, hence we take the second value, Algebraic approach was used to find OP? above, but we can also find OP? by analyzing the diagram.
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-cis(240) (by the way cis(?) means ?? cos(?)+isin(?)) 1. Use graphing software to plot these roots on an Argand diagram as well as a unit circle with centre origin. 1. Choose a root and draw line segments from this root to the other two roots. 1. Measure these line segments and comment on your results. As we see that the length of line segments are equal to each other. (one segment is 0.01 unit bigger than the others because of accuracy of graphing software)
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Handshaking Lemma: For any graph G, the sum of degrees of the vertices in G is twice the size of G. Pigeonhole Principle: n pigeons in m holes, and n>m, then there must be at least one hole containing more than one pigeon. Planar graph: a graph without any edges crossing each other For graph, G, to be planar: If G is a simple graph: If G is a bipartite graph: Any subgraph that contains K5 or K3,3 will not be planar A connected graph has an Eulerian circuit if and only if ALL of its vertices are even.
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Math IA type I. Here is Lacsaps Fractions (the symmetrical pattern given) from n=1 until n=5, again with red numbers representing n:
Here is ?Lacsap?s? Fractions (the symmetrical pattern given) from n=1 until n=5, again with red numbers representing n: 1. 1 1 2. 1 1 3. 1 1 4. 1 1 5. 1 1 If you will notice, row 1 (n=1) has a numerator of 1; row 2 has a numerator of 3; row 3 has a numerator of 6, row 4 has a numerator of 10, and row 5 has a numerator of 15. The numbers of these numerators can be found in the above example of Pascal?s Triangle (column 2 or c2; or c7 but we?ll focus on c2).
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Finally, I will evaluate the benefits and limitations of the models I come up with. Part 1: According to the question, Adam wins twice as many points as Ben does in the game of tennis; therefore, the ratio of points won by Adam to Ben is 2:1 respectively. This shows that Adam wins of the points and Ben wins of the points. The distribution of X, i.e., the number of points won be Adam is derived using the binomial distribution, by substituting the variables and the constants, we are able to arrive at the appropriate model for distribution of X, the number of points won by Adam.
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I am going to go through some logarithm bases, by continuing some sequences and finding an equation to find the nth terms in the sequences
Each subscript will have a raised power and that is the same as the number in the sequence it is or the nth terms. By setting one of the logarithms in the sequence to y with a raised power of n subscript you can change the equation to exponential form and solve for y to find the equation for the nth term. 1. Log28, log48, log88, log168, log328, log648, log1288 , , , , , = y (2n)y= 23 the 2?s will cancel out and give you ny= 3 y= Next to prove this I put into my calculator.
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2x=23, 22x=23, 23x=23, 24x=23, 25x=23, 26x=23, 27x=23 Once I had converted all the bases to be the same I came to see another pattern that was set-up within the sequence and that was that.. 2nx = 23 In this equation n is equal to the nth number. However this equation could be simplified even further due to the fact that the bases are equivalent. nx=3 x=3/n Through this equation you are able to determine the x value through dividing 3 by the nth value.
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In this investigation, I will be modeling the revenue (income) that a firm can expect given it demand curve using my knowledge of linear and quadratic functions
The demand equation shows the relationship between the price of a good (P) and the quantity demanded by consumers (Q), it should be written like this: y = mx + c (straight-line equation). When shown graphically, the price of a good is always shown on the y-axis whereas the quantity demanded always lies on the x-axis. Using our knowledge of linear functions, we can calculate the gradient for each quarter using the market data providing us with information relating to the price and quantity sold.
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Lacsaps fractions are an arrangement of numbers that are symmetrically repeating based on a constant pattern.
+6 +7 +8 As you can see the pattern can then be expressed as where the row number is n, hence I just added 1 to the row number and then added the sum to the previous Numerator and got the Numerator for the subsequent row. Now I will plot the numerator values against the row number in a scatter plot. As you can see above the numerator and the row number exhibit a geometric relationship (due to the shape of the graph).
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Lascap's Fractions. I was able to derive a general statement for both the numerators and denominators, and prove it for other rows of the pattern
Using Microsoft Excel, I plotted the relation between the row number, n, and the numerator in each row. From the graph above, I noticed that the trend line of the relation between the row number and the numerator in each row is quadratic. To represent this graph in a general statement, I used a graphic calculator to derive the general statement using the quadratic regression function, following these steps. 1. Click Stat, then select ?1:Edit? 1. Put the row number in L1 and the numerator of each row on L2 1. Then click STAT, select CALC, then select ?5:QuadReg? 1.
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The purpose of this investigation is to find out how we can scale Barbie to a real life proportion by measuring her height, bust, waist and hips. And then reflect upon these results
The [a]proportion is 1:7.99. [b]Thereafter simply just multiply the proportion by each of Barbie?s measurements (Height, Bust, Hips) and the answers is the measurements for the barbified Libby. The same method is used for the average UK woman where the proportion was 9.7 because the average UK waist size is 86.3cm and 86.3/8.9=9.7. [c] Libby If Libby?s waist remains the same size, then using fractions we see that (Libby?s waist/Barbie?s waist = 71.1 cm/8.9 cm 7.99) Applying that scale factor to the rest of Barbie?s measurement we get the following: Libby?s barbified height: 29.5x7.99 =235.706 236cm Libby?s barbified bust: 11.6 cm x7.99 = 92.684 cm 92.7 cm Libby?s barbified hips: 12.7x 799 = 101.473 cm 101.5 cm[d] Height Bust Waist Hips Barbified Libby (My calculations)
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A constraint in this task is that the ?t? values cannot go before 1896; as the Olympic Games Men?s High Jump did not exist until 1896. Furthermore, there will always be a limit to how high the record will be (h-value) due to the gravitational force of the Earth. Let?s now analyse Graph 1 in order to determine what kind of function best fits the given behaviour of the graph. However, before we proceed, let?s simplify the data points; instead of actual years like 1938 and 1980, which would cause big constant values in the equations we will do later
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The numbers of dots increase by (n+1) adding 2, 3, 4 and 5. Therefore, this hints that the next three terms will be as we will be adding 6, 7, 8. Next Three Terms From the above triangular pattern, we can deduce a general statement which can represent the nth triangular number in terms of n. Butin order to do this, a table of n, Tn,1st difference and 2nd difference should be drawn: n 1 2 3 4 5 6 7 8 Tn 1 3 6 10 15 21 28 36 1st diff.
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The values for the years would have to be rearranged. This is done so that at 1980 x=0, and a y-intercept would occur. Therefore, at x=0, the y-intercept should be 426.8. By acquiring the x and y values through rearranging the values, it would be more convenient later when you substitute the values into the model function to find the coefficients. Table 2: This shows the rearranged years of the 1st section starting with 1980 where x=0. Years Rearranged Years Total Mass (tonnes)
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To find the nth term of this sequence in form of p/q: Log 2 8, Log 4 8, Log 8 8, Log 16 8, Log 32 8, Log 64 8, Log 128 8 The bases can be written as 21, 22, 23 up to 27. Hence the argument being same(8) bases can be generally denoted by 2n where n is the number of term. Hence the logarithm equation for the nth term is : Log 2^n 8 But we have to express in fraction form(p/q form hence)
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number T1 T2 T3 T4 T5 T6 Dots number 1 3 6 10 15 21 I made this table to show that numbers of dots in each term is equal to the number of dots of that term plus the number of the previous term: T1:1+2=3 T2:1+2+3 =6 T3:1+2+3+4=10 T4 :1+2+3+4+5=15. I think it is clear example of arithmetic progression, and to find a general statement I can use the formula from the way we find a sum of certain numbers of an arithmetic sequence: Tn=n(a1+an)/2 where an =a1 +(n-1)d, so I have Tn= n(2a+d(n-1))/2 where a represents the number of dots in the first term, and d is the common difference between terms.
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This explains the vertex of y = x2 + 3 moving three units upwards (the y value changes). The same principle applies to y = x2 ? 2. In this case, two is subtracted from the value of y (relative to y = x2) and therefore the vertex (and the graph) shifts two units downwards (when x = 0, y = -2). Following are all three curves combined into one sketch: From the above observations, we can say that: When |k| units are added to y, the graph shifts upward by k units.
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2 2 3 3 4 4 After subbing different values for and From the above results, by comparing with , it can be seen that their values are opposite, have negative results, ?s results are always a positive number or bigger than 0. A graph of y1 and y2 is shown below when a= 3 and b=5, We know that has zeros , while has opposite concavity to ,which is in the form . From the graph, it can be seen that, is a reflection of , Therefore, the equation of the quadratic is : .
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Thus, the numerator in Figure 1 can be shown as, (n+1)C2 [Eq.1] where n represents row numbers. Sample Calculation - When n=1 (1+1)C2 (2)C2 -When n=2 (2+1)C2 (3)C2 -When n=5 (5+1)C2 (6)C2 15 Caption: The row numbers above are randomly selected within a range of 0?x?5. Therefore, the numerator of 6th row can be found by, (6+1)C2 (7)C2 x = 21 [Eq. 2] and the numerator of 7th row also can be found by, (7+1)C2 (8)C2 x = 28 [Eq.
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... Figure 2: Table of the specific numerators for the first nine rows. Apparently we have a non-linear slope, because the difference between Xn and Xn-1 enlarges with increasing n. Afterwards, the data was plotted in a diagram. Figure 3: Numerator dependent to the specific row shown in a diagram. ________________ It is obvious that the numerators increase non-linear with the row number. This leads to the fact that the values are arranged in a parabolic form. Therefore, it should be possible to find a general formula to describe this data. In order to do so I calculated a line of best fit with a GTR.
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The next form of knowledge used is a modified version of the cos ? rule. The formula for the original cosine rule can be rearranged to be used for finding the angle if all three sides are known. The equation for this rule it cos A=. The last piece of knowledge that will be used is the similar triangle theorem. Similar triangles have the same shape but not the same size. Similar triangles are proportional and have the same value of angles. Investigation: In the first part of this investigation the radius will stay the same and the variable will be the length of OP.
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It was discovered that the Number of sides was equal to four times the previous amount of sides. Nn= Nn-1 x 4 The Length of the original iteration was given as 1 and with each iteration the fractal divides each side by three. So the side length became a third of the previous. Ln = The Perimeter of the Koch snowflake was rather simple too. As the length of each side of the snowflake is the same, the perimeter is equal to the amount of sides multiplied by the length of the sides.
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Drawing a line from A to P? I am able to divide the triangle into two forming another (smaller) isosceles triangle. Now the angles of sides O and P? will be equal, as well as the side lengths of O to A and P? to A. Provided that r=1, I need to find the values of O to P? when OP=2, OP=3 and OP=4. Providing a description to what it is that I have found in my results I need to come up with a general statement for this procedure.
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Guide Number (from tip) 1 2 3 4 5 6 7 8 Distance from Tip (cm) 10 22 34 48 64 81 102 124 How well does your quadratic model fit this new data? What changes, if any, would need to be made for that model to fit this data? Discuss any limitations to your model. Introduction: Fishing rods use guides to control the line as it is being casted, to ensure an efficient cast, and to restrict the line from tangling. An efficient fishing rod will use multiple, strategically placed guides to maximize its functionality.
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