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Maths Driving test data: analysis and manipulation.
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Introduction The objective of this project is to successfully manipulate a large set of data in order to prove/ disprove a set of hypotheses. The data consists of results and statistics from a set of driving instruction and tests. The dependant variable in the data is the number of mistakes made during the test. The hypotheses will have been synthesized by myself and in response to influenced by the data that I have been given. The data will have to be subjected to sampling to reduce the vast amount of information. The data will then be processed so...

M 35 5 C Wed 14

M 40 4 C Thur 13

F 10 37 D Thur 10

F 17 31 D Wed 13

F 24 28 D Mon 10

F 31 24 D Mon 17

F 32 17 D Fri 14

F 40 19 D Fri 17

M 15 35 D Wed 14

M 25 28 D Thur 9

M 28 26 D Tue 12

M 29 27 D Thur 12

M 40 4 D Fri 14

M 40 20 D Wed 16

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Introduction In this investigation I...Introduction In this investigation I have been given the task of discovering a formula to tell us the value of n when 'n' is any given number. Also, in the second part of my investigation, look into whether there is a link between the phi of a number, and the product of its factors' phis. Such as: Is it true that 6 x 4 = 6 x 4? How to Find the Phi of a Number For any positive integer n, the phi function n is defined as the number of positive integers les than n which have no factor other than 1 in common are co-prime with n. So 10 = 4 because the positive integers less than 10 which have no factors other than 1, in common with 10 are 1, 3, 7 and 9 i.e. 4 of them. Also 16 = 8 because the integers less than 16 which have no factors other than 1, in common with 16 are 1, 3, 5, 7, 9, 11, 13 and 15 i.e. 8 of them. Part 1 i 3 = 1, 2 = 2 ii 8 = 1, 3, 5, 7 = 4 iii 24 = 1, 5, 7, 9, 11, 13, 17, 19, 24 = 9 iv 11 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10 To investigate this idea further, I will now find the phi functions for the numbers up to, and including 20, to see if I can find any patterns which will indicate to me a formula for n. ** In this investigation if the phi of any number has already been studied, only the phi will be given, without any working. ** ** Also only the numbers which are not factors of the phi in question will be stated. ** 1 = 0 2 = 1 = 1 3 = 1, 2 = 2 4 = 1, 3 = 2 5 = 1, 2, 3, 4 = 4 6 = 1, 5 = 2 7 = 1, 2, 3, 4, 5, 6 = 6 8 = 1, 3, 5, 7 = 4 9 = 1, 2, 4, 5, 7, 8 = 6 10 = 1, 3, 7, 9 = 4 11 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10 12 = 1, 5, 7, 11 = 4 13 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 = 12 14 = 1, 3, 5, 9, 11, 13 = 6 15 = 1, 2, 4, 7, 8, 11, 13, 14 = 8 16 = 1, 3, 5, 7, 9, 11, 13, 15 = 8 17 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 16 18 = 1, 5, 7, 11, 13, 17 = 6 19 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 = 18 20 = 1, 3, 7, 9, 11, 13, 17, 19 = 8 Sequence: 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8. Summary From studying the phi of numbers 1-20 there is clearly no outstanding link between the numbers, and their phi's. Although I can draw several points from this: Prime Numbers The prime numbers from the previous sequence were: 2, 3, 5, 7, 11, 13, 17, and 19. The phis of these were: 2 = 1 3 = 2 5 = 4 7 = 6 11 = 10 13 = 12 17 = 16 From this it can be seen that the phi of a prime number is always one less than the number in question e.g. 17 = 16. This is because there a no factors which go into a prime number, except for the number 1 and itself, and in the phi function the number in question is not considered, and the number 1 is always a factor, therefore all numbers except for the number in question, are all counted. We can now draw from this a simple formula where 'n' is any given PRIME number: n = n-1 Odd or Even? All of the results were even, except for the phi of 2 which is 1. Factors within a Phi Other than Prime numbers themselves and numbers which have factors which are prime numbers, have a phi of how many prime numbers exist in them, such as the phi of 8: 8 = 1, 3, 5, 7 In this phi only the numbers: 1, 3, 5 and 7 do not go into the number 8, and these numbers are all prime, except for 2 which is a factor of all even numbers therefore the phi of 8 has the same number of prime numbers in it, as the value of its phi, which is 4. This pattern also exists in the phis of: 3 4 8 already stated 10 14 16 The Phi's of Powers of Prime Number How to work out the Phi's of Powers of Prime Number The phi of a power of a prime number can still be found by identifying the number of positive integers less than n, which have no factor other than 1 in common are co-prime with n, but a simple amount of BIDMASS must be employed. This means that the phi of a power of a prime number can be found but multiplying the brackets out first, must occur, and then the phi of that number can be found. Below the first 3 prime numbers in the sequence of prime numbers 2, 3, 5, 7"¦.. and there first 5 power numbers and their phis have been found. Powers of Prime Numbers and their Phi's ** In this next part of my study the sign ^ will indicate a powered number, such as 2^2, 2^3, 2^4"¦"¦.. ** 2 = 1 2^2 = 2 2^3 = 4 2^4 = 8 2^5 = 16 3 = 2 3^2 = 6 3^3 = 18 3^4 = 54 3^5 = 162 5 = 4 5^2 = 20 5^3 = 100 5^4 = 500 5^5 = 2500 From the prime numbers and their powers phi's it can be seen that a clear pattern has become apparent. It can be seen that each time the power goes up by 1, the phi of that powered prime, is the previous powered prime's phi, multiplied by the prime number in question. I know this because it is evident in the previously stated phi's, for example from the powers of 2 phi functions, 1 x 2 is 2, 2 x 2 is 4, 4 x 2 is 8 and so on.   

Introduction In this investigation I have been given the task of discovering a formula to tell us the value of n when 'n' is any given number. Also, in the second part of my investigation, look into whether there is a link between the phi of a number,...

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Minimum Surface Area Investigation Introduction...Minimum Surface Area Investigation Introduction For my maths coursework I have been asked by a tin manufacturing company to minimise the amount of metal required to produce cans. The cans must be cylindrical and come in four different sizes: 1. 300 cm3 2. 400 cm3 3. 600 cm3 4. 750 cm3 I have been asked to come up with a reverse formula to enable them to find the height and radius for any given formula of a cylindrical tin that will ensure that the surface area is a minimum. Method I am going to devise a method using a spreadsheet package called excel. I have decided to use a spreadsheet because it is accurate, easy to use and any adjustments can be easily made. I can see immediate results and I can produce graphs of all kinds. The spreadsheet will use formulae which will calculate the surface areas, heights and radii by just entering key information. These formulae can be easily adjusted or corrected. I will use this spreadsheet to calculate the four given formula's: 1. 300 cm3 2. 400 cm3 3. 600 cm3 4. 750 cm3 I will get a greater degree of accuracy in my results by adjusting the radius from initially an integer then to decimal of 1 decimal place, then 2 decimal places, then 3 decimal places. This will bring all the results to a greater accuracy. The spreadsheet will use formulas to make it more efficient. The formulae in the spread sheet will be The Formulae I Used"¦ Area of side of cylinder = c x h Volume = 2πr3h The total surface area of the cylinder = 2πr r+h Height = _V_ πr2 Table Of Results For Cylinder "“ Volume 300cm3 Radiuscm Heightcm Volumecm3 Surface Areacm3 1 95.4930 300.00 606.283185307 2 23.8732 300.00 325.132741229 3 10.6103 300.00 256.548667765 4 5.9683 300.00 250.530964915 5 3.8197 300.00 277.079632679 6 2.6526 300.00 326.194671058 7 1.9488 300.00 393.590365766 8 1.4921 300.00 477.123859659 9 1.1789 300.00 575.604676548 10 0.9549 300.00 688.318530718 3.0 10.6103 300.00 256.548667765 3.1 9.9368 300.00 253.929797899 3.2 9.3255 300.00 251.839817546 3.3 8.7689 300.00 250.242069813 3.4 8.2606 300.00 249.104210386 3.5 7.7953 300.00 248.397591442 3.6 7.3683 300.00 248.096748248 3.7 6.9754 300.00 248.178969017 3.8 6.6131 300.00 248.623932678 3.9 6.2783 300.00 249.413402368 4.0 5.9683 300.00 250.530964915 3.60 7.3683 300.00 248.096748248 3.61 7.3275 300.00 248.088085391 3.62 7.2871 300.00 248.083229893 3.63 7.2470 300.00 248.082160673 3.64 7.2072 300.00 248.084856881 3.65 7.1678 300.00 248.091297899 3.66 7.1287 300.00 248.101463330 3.67 7.0899 300.00 248.115333003 3.68 7.0514 300.00 248.132886965 3.69 7.0132 300.00 248.154105477 3.70 6.9754 300.00 248.178969017 3.620 7.2871 300.00 248.083229893 3.621 7.2831 300.00 248.082952934 3.622 7.2790 300.00 248.082713817 3.623 7.2750 300.00 248.082512521 3.624 7.2710 300.00 248.082349024 3.625 7.2670 300.00 248.082223306 3.626 7.2630 300.00 248.082135347 3.627 7.2590 300.00 248.082085124 3.628 7.2550 300.00 248.082072618 3.629 7.2510 300.00 248.082097808 3.630 7.2470 300.00 248.082160673 Values For Minimum Surface Area, radius & Height. Table Of Results For Cylinder "“ Volume 400cm3 Radiuscm Heightcm Volumecm3 Surface Areacm3 1 127.3240 400.00 806.283185307 2 31.8310 400.00 425.132741229 3 14.1471 400.00 323.215334431 4 7.9577 400.00 300.530964915 5 5.0930 400.00 317.079632679 6 3.5368 400.00 359.528004392 7 2.5984 400.00 422.161794338 8 1.9894 400.00 502.123859659 9 1.5719 400.00 597.826898770 10 1.2732 400.00 708.318530718 3.5 10.3938 400.00 305.540448584 3.6 9.8244 400.00 303.652303803 3.7 9.3005 400.00 302.233023072 3.8 8.8174 400.00 301.255511625 3.9 8.3711 400.00 300.695453650 4.0 7.9577 400.00 300.530964915 4.1 7.5743 400.00 300.742296233 4.2 7.2179 400.00 301.311579295 4.3 6.8861 400.00 302.222607958 4.4 6.5767 400.00 303.460649365 4.5 6.2876 400.00 305.012280248 3.92 8.2859 400.00 300.631571357 3.93 8.2438 400.00 300.605509718 3.94 8.2020 400.00 300.583340714 3.95 8.1605 400.00 300.565044325 3.96 8.1193 400.00 300.550600733 3.97 8.0785 400.00 300.539990321 3.98 8.0379 400.00 300.533193665 3.99 7.9977 400.00 300.530191542 4.00 7.9577 400.00 300.530964915 4.01 7.9181 400.00 300.535494941 4.02 7.8788 400.00 300.543762963 3.990 7.9977 400.00 300.530191542 3.991 7.9937 400.00 300.530099294 3.992 7.9897 400.00 300.530044782 3.993 7.9857 400.00 300.530027988 3.994 7.9817 400.00 300.530048891 3.995 7.9777 400.00 300.530107473 3.996 7.9737 400.00 300.530203716 3.997 7.9697 400.00 300.530337601 3.998 7.9657 400.00 300.530509108 3.999 7.9617 400.00 300.530718219 Values For Minimum Surface Area, radius and height. Table Of Results For Cylinder "“ Volume 600cm3 Radiuscm Heightcm Volumecm3 Surface Areacm3 1 190.9859 600.00 1206.283185307 2 47.7465 600.00 625.132741229 3 21.2207 600.00 456.548667765 4 11.9366 600.00 400.530964915 5 7.6394 600.00 397.079632679 6 5.3052 600.00 426.194671058 7 3.8977 600.00 479.304651480 8 2.9842 600.00 552.123859659 9 2.3579 600.00 642.271343215 10 1.9099 600.00 748.318530718 4.0 11.9366 600.00 400.530964915 4.1 11.3614 600.00 398.303271843 4.2 10.8269 600.00 396.549674533 4.3 10.3291 600.00 395.245863772 4.4 9.8650 600.00 394.369740274 4.5 9.4314 600.00 393.901169137 4.6 9.0258 600.00 393.821766317 4.7 8.6458 600.00 394.114712372 4.8 8.2893 600.00 394.764589477 4.9 7.9544 600.00 395.757238409 5.0 7.6394 600.00 397.079632679 4.50 9.4314 600.00 393.901169137 4.51 9.3896 600.00 393.876005493 4.52 9.3481 600.00 393.854714764 4.53 9.3069 600.00 393.837279622 4.54 9.2659 600.00 393.823682894 4.55 9.2253 600.00 393.813907558 4.56 9.1848 600.00 393.807936740 4.57 9.1447 600.00 393.805753715 4.58 9.1048 600.00 393.807341902 4.59 9.0652 600.00 393.812684867 4.60 9.0258 600.00 393.821766317 4.563 9.1728 600.00 393.806884872 4.564 9.1687 600.00 393.806609927 4.565 9.1647 600.00 393.806372793 4.566 9.1607 600.00 393.806173454 4.567 9.1567 600.00 393.806011893 4.568 9.1527 600.00 393.805888094 4.569 9.1487 600.00 393.805802040 4.570 9.1447 600.00 393.805753715 4.571 9.1407 600.00 393.805743101 4.572 9.1367 600.00 393.805770183 4.573 9.1327 600.00 393.805834944 Values For Minimum Surface Area, Radius & Height. Table Of Results For Cylinder "“ Volume 750cm3 Radiuscm Heightcm Volumecm3 Surface Areacm3 1 238.7324 750.00 1506.283185307 2 59.6831 750.00 775.132741229 3 26.5258 750.00 556.548667765 4 14.9208 750.00 475.530964915 5 9.5493 750.00 457.079632679 6 6.6315 750.00 476.194671058 7 4.8721 750.00 522.161794338 8 3.7302 750.00 589.623859659 9 2.9473 750.00 675.604676548 10 2.3873 750.00 778.318530718 4.0 14.9208 750.00 475.530964915 4.1 14.2018 750.00 471.474003550 4.2 13.5336 750.00 467.978245962 4.3 12.9114 750.00 465.013305632 4.4 12.3312 750.00 462.551558456 4.5 11.7893 750.00 460.567835804 4.6 11.2823 750.00 459.039157622 4.7 10.8073 750.00 457.944499606 4.8 10.3616 750.00 457.264589477 4.9 9.9430 750.00 456.981728205 5.0 9.5493 750.00 457.079632679 4.85 10.1491 750.00 457.074576904 4.86 10.1074 750.00 457.048298990 4.87 10.0659 750.00 457.025891164 4.88 10.0247 750.00 457.007337360 4.89 9.9837 750.00 456.992621641 4.90 9.9430 750.00 456.981728205 4.91 9.9026 750.00 456.974641374 4.92 9.8624 750.00 456.971345600 4.93 9.8224 750.00 456.971825461 4.94 9.7827 750.00 456.976065659 4.95 9.7432 750.00 456.984051019 4.920 9.8624 750.00 456.971345600 4.921 9.8584 750.00 456.971223936 4.922 9.8544 750.00 456.971140013 4.923 9.8504 750.00 456.971093815 4.924 9.8464 750.00 456.971085328 4.925 9.8424 750.00 456.971114535 4.926 9.8384 750.00 456.971181422 4.927 9.8344 750.00 456.971285974 4.928 9.8304 750.00 456.971428174 4.929 9.8264 750.00 456.971608008 4.930 9.8224 750.00 456.971825461 Values For Minimum Surface Area, Radius & Height. Explanation Of Why The Height Of A Cylinder Is Twice The Radius When The Surface Area Is At A Minimum. The most efficient three dimensional shape to contain a volume is a sphere, so the cylinder tries to imitate a sphere in order to achieve the minimum surface area. So as further investigation for the tin manufacturing company, I will now look at alternate minimum shapes & therefore reduce expenditure. I'll now calculate the minimum surface area of a sphere & a cube with a volume of 900cm3 in addition to the cylinder I have already correctly predicted. This further investigation will tell me whether a cylinder is the best 3D shape to use. Sphere V = 4_ πr3 S.A = 4 πr2 3 900 = 4_ πr3 S.A = 4π x 5.989418137 2 3 2700 = 4 πr3 S.A. = 450.7950419 2700 = r3 4π r = 3√2700 4π r = 5.989418137 Cube Volume = l3 900 = l3 l = 3√900 l = 9.654893846 S.A. = 6l3 = 6 x 9.6548938462 S.A. = 559.3018511 Analysis Of Investigation into which Spreadsheet To Use After Investigating the minimum surface Area for a volume of 900cm3 my results were: "¢ Cylinder "“ 516.0315049 cm3 "¢ Sphere - 450.7950419 cm3 "¢ Cube - 559.3018511 cm3 So from these results I can see that as I previously stated the sphere is the best 3-D shape to store any volume water because it has the smallest surface area. However the manufacturing company has chosen the cylinder because although a sphere has a smaller surface area it would be very difficult to stack on supermarket shelves and store safely, whereas a cylinder can be kept upright.   

Minimum Surface Area Investigation Introduction For my maths coursework I have been asked by a tin manufacturing company to minimise the amount of metal required to produce cans. The cans must be cylindrical and come in four different sizes: 1. 300 cm3 2. 400 cm3 ...

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