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estudier Group Title

Piece of wood width 1 sits over a square tile through opposite corners (see pic) What proportion of the area of the tile is covered by the wood?

  • one year ago
  • one year ago

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  1. estudier Group Title
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    |dw:1350379170293:dw|

    • one year ago
  2. estudier Group Title
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    Bad drawing, u probably should do your own.....:-)

    • one year ago
  3. estudier Group Title
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    If u can see it opposite edges of the wood are aligned with corners of tile.

    • one year ago
  4. JamesWolf Group Title
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    |dw:1350379418357:dw|

    • one year ago
  5. estudier Group Title
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    @JamesWolf U should be practicing your Python!:-)

    • one year ago
  6. estudier Group Title
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    That pic is not correct, the edges are aligned with the corners, not in the middle...

    • one year ago
  7. JamesWolf Group Title
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    so a is the angle between the middle of the wood and the side of the square which is 90/2 = 45. the bottom of that triangle has length 1/2. we need to find b. then you can do "length of square" - b = side of triangle|dw:1350379684247:dw|

    • one year ago
  8. JamesWolf Group Title
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    righttt sorry

    • one year ago
  9. JamesWolf Group Title
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    opposite corners, yes i should be doing python! stupid python

    • one year ago
  10. estudier Group Title
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    :-)

    • one year ago
  11. JamesWolf Group Title
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    i think |dw:1350379781386:dw| is the same as |dw:1350379796776:dw|

    • one year ago
  12. estudier Group Title
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    |dw:1350379795901:dw|

    • one year ago
  13. JamesWolf Group Title
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    asin, there is the same amount of mising area

    • one year ago
  14. sara12345 Group Title
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    |dw:1350379782354:dw|

    • one year ago
  15. sara12345 Group Title
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    like this ?

    • one year ago
  16. estudier Group Title
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    @sara12345 Yes, like that

    • one year ago
  17. JamesWolf Group Title
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    |dw:1350379878261:dw| i think the two diagrams are equivilant

    • one year ago
  18. sara12345 Group Title
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    |dw:1350379928886:dw|

    • one year ago
  19. estudier Group Title
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    @JamesWolf I don't mind how u do it, your choice....

    • one year ago
  20. estudier Group Title
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    @sara12345 What r u going to do about the little corner bits?

    • one year ago
  21. JamesWolf Group Title
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    how about i do it my way, and sara can do it her way, and we will both solve for area. and see if they really are equivilant?

    • one year ago
  22. estudier Group Title
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    I will tell u there is there is also a quick way to do it but I will give medal for correct answer by any method.

    • one year ago
  23. JamesWolf Group Title
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    \[A = \frac{1}{2}(l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]

    • one year ago
  24. JamesWolf Group Title
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    that formula makes me think im wrong lol

    • one year ago
  25. JamesWolf Group Title
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    whats your method estudier?

    • one year ago
  26. estudier Group Title
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    Let's wait, let people have a go first.....

    • one year ago
  27. JamesWolf Group Title
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    \[A = (\frac{1}{2}l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]

    • one year ago
  28. JamesWolf Group Title
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    alright so for a square with side length 5, i get - 14. lol

    • one year ago
  29. estudier Group Title
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    I haven't done it with figures, let me check....

    • one year ago
  30. JamesWolf Group Title
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    try with side length 5

    • one year ago
  31. estudier Group Title
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    I get a proportion of 1/4

    • one year ago
  32. JamesWolf Group Title
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    of width of wood to area?

    • one year ago
  33. estudier Group Title
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    Proportion of area covered by wood

    • one year ago
  34. JamesWolf Group Title
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    i get 1/3.888 you got exactly 1/4?

    • one year ago
  35. estudier Group Title
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    |dw:1350381191180:dw|

    • one year ago
  36. JamesWolf Group Title
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    \[2(\frac{1}{2} \times (l - \sqrt(2))^2)\] area not covered.

    • one year ago
  37. estudier Group Title
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    OK, unless someone comes with answer in a minute, I will post answer...

    • one year ago
  38. JamesWolf Group Title
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    :)

    • one year ago
  39. estudier Group Title
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    |dw:1350381831150:dw|

    • one year ago
  40. sauravshakya Group Title
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    A minute plz

    • one year ago
  41. estudier Group Title
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    K

    • one year ago
  42. JamesWolf Group Title
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    i think i see what your going to do!

    • one year ago
  43. JamesWolf Group Title
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    estudier that is.

    • one year ago
  44. JamesWolf Group Title
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    hmm, not sure how your going to get XC

    • one year ago
  45. JamesWolf Group Title
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    if thats what your going to do...

    • one year ago
  46. estudier Group Title
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    A longer way is to first see that APX and XBC are similar...

    • one year ago
  47. JamesWolf Group Title
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    Im not very good at geometry so im not sure how that helps

    • one year ago
  48. estudier Group Title
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    But a quicker way is to note that the wood area is a //gram with base x and height a

    • one year ago
  49. estudier Group Title
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    It is also a //gram with base XC and height 1

    • one year ago
  50. estudier Group Title
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    So xa = sqrt((x-a)^2 +a^2) Square and simplify gives you a quadratic...

    • one year ago
  51. JamesWolf Group Title
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    sorry, what is a?

    • one year ago
  52. JamesWolf Group Title
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    a = A

    • one year ago
  53. estudier Group Title
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    a is side of square, you changed it to l

    • one year ago
  54. sara12345 Group Title
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    side of square

    • one year ago
  55. JamesWolf Group Title
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    oh right sorry lol

    • one year ago
  56. JamesWolf Group Title
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    carry on

    • one year ago
  57. estudier Group Title
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    You can get the same quadratic by similar triangles as I said before: sqrt(x^2-1)/1 = (a-x)/a

    • one year ago
  58. estudier Group Title
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    Solving the quadratic and calculating the proportion gives 2/ (sqrt(2a^2-1) + 1)

    • one year ago
  59. sara12345 Group Title
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    we wil get x/a = 1/4 ?

    • one year ago
  60. estudier Group Title
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    That was for a = 5 only....

    • one year ago
  61. sara12345 Group Title
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    if i go ahead and solve the quadratic, ilg et x/ = 2/ (sqrt(2a^2-1) + 1) ?

    • one year ago
  62. sara12345 Group Title
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    x/a = 2/ (sqrt(2a^2-1) + 1)

    • one year ago
  63. estudier Group Title
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    No, you will get x = (an expression) but we want a proportion so multiply by a and divide by a^2

    • one year ago
  64. JamesWolf Group Title
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    sorry as i said im rubbish, can you explain what side each term is refering too in this So xa = sqrt((x-a)^2 +a^2)

    • one year ago
  65. estudier Group Title
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    Both sides are the area of the parallelogram (the wood area) Base * "height"

    • one year ago
  66. JamesWolf Group Title
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    or just write on here :) |dw:1350382973068:dw|

    • one year ago
  67. sara12345 Group Title
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    |dw:1350383010964:dw|

    • one year ago
  68. sara12345 Group Title
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    for that ||gram, if we see bottom as base, then height = a

    • one year ago
  69. JamesWolf Group Title
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    so sqrt((x-a)^2 +a^2) is working out the length from X to C

    • one year ago
  70. sara12345 Group Title
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    correct

    • one year ago
  71. sara12345 Group Title
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    then height =1

    • one year ago
  72. JamesWolf Group Title
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    okay I see thanks

    • one year ago
  73. estudier Group Title
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    Good geometry practice:-)

    • one year ago
  74. sara12345 Group Title
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    as of now im not getting how to simplify quadratic formula.. il work later

    • one year ago
  75. sara12345 Group Title
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    yesss

    • one year ago
  76. JamesWolf Group Title
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    back to python :)

    • one year ago
  77. estudier Group Title
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    Hang on I will put the quadratic.....

    • one year ago
  78. sara12345 Group Title
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    i got : x^2(1-a^2) + 2ax + 2a^2 = 0

    • one year ago
  79. sara12345 Group Title
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    x^2(1-a^2) - 2ax + 2a^2 = 0

    • one year ago
  80. estudier Group Title
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    x = a(-1 + sqrt(2a^2-1) ) / ( a^2-1) (dropping the - soln) Multiply by a, divide by a^2 -> (sqrt(2a^2-1) -1) / (a^2-1) -> equivalent what I gave before...

    • one year ago
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