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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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Hidden Faces Coursework A cube...Hidden Faces Coursework A cube a total of 6 sides, when it is places on a surface only 5 of the 6 faces can be seen. However if you place 5 cubes side by side, there is a total of 30 faces, but out of this 30 only 17 can be seen. In this coursework I will be finding out the global formula for the total number of hidden faces for any number of cubes in any way positioned. To find this out I will be testing various numbers of cubes in different positions. This will enable me to find out several different formulae. Using the formulas found I will then be able to find out the global formula. I am generating only 3 formulae to get to the global formula. 1 row 6 faces 1 cube 1 hidden face 1 row 12 faces 2 cubes 4 hidden faces 1 row 18 faces 3 cubes 7 hidden faces 1 row 24 faces 4 cubes 10 hidden faces 1 row 30 faces 5 cubes 13 hidden faces 1 row 36 faces 6 cubes 16 hidden faces 1 row 42 faces 7 cubes 19 hidden faces 1 row 48 faces 8 cubes 22 hidden faces From the cubes drawn above I can see a pattern being formed. The number of hidden faces goes up by 3 every time a cube is added on the end. Cubes in a row Total faces Faces seen Faces unseen 1x1 6 5 1 1x2 12 8 4 1x3 18 11 7 1x4 24 14 10 1x5 30 17 13 1x6 36 20 20 1x7 42 23 19 1x8 48 26 22 The graph above show the number of hidden faces, the number of faces which can be seen and the total number of faces. Nth term 1 2 3 4 5 6 7 8 Total faces 6 12 18 24 30 36 42 48 difference + 6 + 6 + 6 + 6 + 6 + 6 + 6 The table above shows the total number of faces on an 'n' number of cubes. As we increase the number of cubes being added on the number of faces increases by 6. The formula to find out the total number of faces is: 6n E.g. 4 is the nth term so you multiply 4 by 6, which gives you a total of 24 which is the answer. Nth term 1 2 3 4 5 6 7 8 Seen faces 5 8 11 14 17 20 23 26 difference + 3 + 3 + 3 + 3 + 3 + 3 + 3 The table above shows the amount of faces seen on an 'n' number of cubes. The formula for working out the number of faces which can be seen is: 3n+2 E.g. 3 is the nth term so you have to multiply 3 by 3 33+2 Which gives a total of 9. you then add 2 which gives a final total of 11. Below shows the relationship between the cubes and the number of faces. Both hidden and seen. Nth term 1 2 3 4 5 6 7 8 Hidden faces 1 4 7 10 13 16 19 22 differences + 3 + 3 + 3 + 3 + 3 + 3 + 3 The graph above shows how many hidden faces there are related to the number of cubes. The graph and the table above shows the relationship between the number of cubes and the number of faces seen and unseen. Both the graph and the table above will now allow me to work out the formula for the number of hidden faces in one row. To find the global formula for the number of hidden faces in one row I have to refer to the table above. As you can see from the table it will be a linear equation because there is only 1 line of difference. The general linear equation is y=mx+c Therefore the linear rule is in the form of tn=an+c In the equation tn is the total number of hidden faces and n is the number of cubes. Therefore I need to find out the equation for a and c are. In the equation a is equal to the first difference. So I can replace the a with a 3, which makes tn=3n+c Now I need to find out the value of c so I can substitute it into the equation. To find c I will chose a number of cubes from the table and its results and place it into the equation. e.g. tn=an+c 13=35+c 13=15+c Now all I have to do is rearrange the formula so I can find out c. 13=35+c 13-15=c c=-2 tn=3n+-2 the equation is not in its simplest form so now I need to multiply out the brackets so I can get my final formula. tn=3n-2 To see whether or not the formula works I will test it using a number of cubes which Is not in my table above. I will use 10 cubes for this. tn=an+c tn=3n-2 tn=310-2 tn=30-2 tn=28 As I can see from the above equation I found out that my formula for hidden faces in one row has worked. For the next part of my coursework I need to generate another formula but for a different structure of cubes. I will be using 2 rows for this part of the coursework. 2 rows 2 cubes 12 faces 4 hidden Faces 2 rows 4 cubes 24 faces 12 hidden faces 2 rows 6 cubes 36 faces 20 hidden faces Cubes In A Row Total Faces Seen Faces Hidden Faces 2x1 12 8 4 2x2 24 12 12 2x3 36 16 20 2x4 48 20 28 2x5 60 24 36 2x6 72 28 44 2x7 84 32 52 2x8 96 36 60 12n 4n+4 8n-4 The graph above shows us the number of cubes, faces a hidden faces. It also shows the formulae for finding out the number of hidden faces, total faces and number of faces seen. Above shows the relationship between the number of cubes and the number of faces. By looking at the graph and the chart I have generated my second equation to find the number of hidden faces in 2 rows. The table below shows the relationship between the number of cubes and the number of hidden faces. I did not draw the other two tables because they were not relevant in finding out the global formula. nth term 2 4 6 8 10 12 14 16 tn 4 12 20 28 36 44 52 60 1st diff + 8 + 8 + 8 + 8 + 8 + 8 + 8 Formula for working out the total number of hidden faces: This pattern also has only the 1st difference so I could see that this was going to be a linear equation as well. So again I would have to follow the rule: tn = an + c As you know tn is total number of hidden faces and n is the number of cubes, so the two unknowns are once again a and c. a has been replaced with 8 as that represents the 1st difference, therefore the equation now looks like: tn = 8n+c. Again two methods can be used to work out the c term. The first way, which can be used, is to use the zero term: so as the 1st difference is +3 the working out would be 4 "“ 8 = -4. Another method that could be used is, you pick any number from the table above and it's result, and work it out as an equation: tn = 8n+c 44 = 86+c 44 = 48+c You need to rearrange the formula so c is on it's own, and change the + to -. c = 44 "“ 48 c = -4 So the second general formula would look like: tn = 8n - 4 To make sure my formula is right I will test it but I will use a value of cubes which is not in my table. I will use 20 cubes. Now all I have to do is substitute the information found into the equation. The final part of my investigation is to generate another formula, all these three formulae should help me find out the global formula. To get my third and final formula I will investigate using 3 rows of cubes. This should hopefully help me get the global formula. Cubes In A Row Total Faces Seen Faces Hidden Faces 3x1 18 11 7 3x2 36 16 20 3x3 54 21 33 3x4 72 26 46 3x5 90 31 59 3x6 108 36 72 3x7 126 41 85 3x8 144 46 98 18n 5n+6 13n-6 The table above shows the formulae for total faces, hidden faces and seen faces. The graph above show the relationship between the faces and the number of cubes. nth term 3 6 9 12 15 18 21 24 tn 7 20 33 46 59 72 85 98 1st diff + 13 + 13 + 13 + 13 + 13 + 13 + 13 The table above shows the difference between the number of hidden faces and the number of cubes. I have to do all the things listed above in the previous investigations, but using the information gained in this investigation. I have to substitute the letters with numbers. To make sure my formula works I will have to test it out first. I will use a number which is not part of my table. I will use 30 cubes. tn=an+c tn=13n-6 tn=1330-6 tn=390-6 tn=384 The final formula is 13n-6 The final section of my coursework is to find out the global formula for finding out the total number of hidden faces in any given number of cubes and rows. I know that the formulae for finding out the different surfaces areas are: length multiplied by width, length multiplied by height and height multiplied by width. I also know the 6LWH will give me the volume of the cube, which is the total number of faces in a group of cubes. This information will help me find out the global formula. As 6LWH is the total number of faces, this can be used as the first part of my formula. 6LWH Now I need to subtract the faces which are not inside of the cube, to do this I will use the formulae for finding out the surface area. Firstly I will have to minus the top are of the cubes as these are all visible, the formula for the surface area is length multiplied by width. This makes our formula 6LWH-LW Now I have to minus the four surface areas that are on the side of the cuboids. two of these will always be equal, as will the other two sides. To find out the surface area of the two different sized sides, I will have to do length multiplied by width for one and width multiplied by height for the other. But because these surfaces are in pairs I will have to multiply them both by two. This will give me the final formula. 6LWH-2HW+2HL-LW The formula shown above is used to work out hidden faces in any number of cubes in any formation. This concludes my coursework and I reached my target, which was to find the global formula by only using three formulae.   

Hidden Faces Coursework A cube a total of 6 sides, when it is places on a surface only 5 of the 6 faces can be seen. However if you place 5 cubes side by side, there is a total of 30 faces, but out of this 30 only 17...

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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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