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estudier

  • 2 years ago

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?

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  1. estudier
    • 2 years ago
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    |dw:1350379170293:dw|

  2. estudier
    • 2 years ago
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    Bad drawing, u probably should do your own.....:-)

  3. estudier
    • 2 years ago
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    If u can see it opposite edges of the wood are aligned with corners of tile.

  4. JamesWolf
    • 2 years ago
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    |dw:1350379418357:dw|

  5. estudier
    • 2 years ago
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    @JamesWolf U should be practicing your Python!:-)

  6. estudier
    • 2 years ago
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    That pic is not correct, the edges are aligned with the corners, not in the middle...

  7. JamesWolf
    • 2 years ago
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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|

  8. JamesWolf
    • 2 years ago
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    righttt sorry

  9. JamesWolf
    • 2 years ago
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    opposite corners, yes i should be doing python! stupid python

  10. estudier
    • 2 years ago
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    :-)

  11. JamesWolf
    • 2 years ago
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    i think |dw:1350379781386:dw| is the same as |dw:1350379796776:dw|

  12. estudier
    • 2 years ago
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    |dw:1350379795901:dw|

  13. JamesWolf
    • 2 years ago
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    asin, there is the same amount of mising area

  14. sara12345
    • 2 years ago
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    |dw:1350379782354:dw|

  15. sara12345
    • 2 years ago
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    like this ?

  16. estudier
    • 2 years ago
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    @sara12345 Yes, like that

  17. JamesWolf
    • 2 years ago
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    |dw:1350379878261:dw| i think the two diagrams are equivilant

  18. sara12345
    • 2 years ago
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    |dw:1350379928886:dw|

  19. estudier
    • 2 years ago
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    @JamesWolf I don't mind how u do it, your choice....

  20. estudier
    • 2 years ago
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    @sara12345 What r u going to do about the little corner bits?

  21. JamesWolf
    • 2 years ago
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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?

  22. estudier
    • 2 years ago
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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.

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

  24. JamesWolf
    • 2 years ago
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    that formula makes me think im wrong lol

  25. JamesWolf
    • 2 years ago
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    whats your method estudier?

  26. estudier
    • 2 years ago
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    Let's wait, let people have a go first.....

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

  28. JamesWolf
    • 2 years ago
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    alright so for a square with side length 5, i get - 14. lol

  29. estudier
    • 2 years ago
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    I haven't done it with figures, let me check....

  30. JamesWolf
    • 2 years ago
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    try with side length 5

  31. estudier
    • 2 years ago
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    I get a proportion of 1/4

  32. JamesWolf
    • 2 years ago
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    of width of wood to area?

  33. estudier
    • 2 years ago
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    Proportion of area covered by wood

  34. JamesWolf
    • 2 years ago
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    i get 1/3.888 you got exactly 1/4?

  35. estudier
    • 2 years ago
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    |dw:1350381191180:dw|

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

  37. estudier
    • 2 years ago
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    OK, unless someone comes with answer in a minute, I will post answer...

  38. JamesWolf
    • 2 years ago
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    :)

  39. estudier
    • 2 years ago
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    |dw:1350381831150:dw|

  40. sauravshakya
    • 2 years ago
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    A minute plz

  41. estudier
    • 2 years ago
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    K

  42. JamesWolf
    • 2 years ago
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    i think i see what your going to do!

  43. JamesWolf
    • 2 years ago
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    estudier that is.

  44. JamesWolf
    • 2 years ago
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    hmm, not sure how your going to get XC

  45. JamesWolf
    • 2 years ago
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    if thats what your going to do...

  46. estudier
    • 2 years ago
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    A longer way is to first see that APX and XBC are similar...

  47. JamesWolf
    • 2 years ago
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    Im not very good at geometry so im not sure how that helps

  48. estudier
    • 2 years ago
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    But a quicker way is to note that the wood area is a //gram with base x and height a

  49. estudier
    • 2 years ago
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    It is also a //gram with base XC and height 1

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

  51. JamesWolf
    • 2 years ago
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    sorry, what is a?

  52. JamesWolf
    • 2 years ago
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    a = A

  53. estudier
    • 2 years ago
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    a is side of square, you changed it to l

  54. sara12345
    • 2 years ago
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    side of square

  55. JamesWolf
    • 2 years ago
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    oh right sorry lol

  56. JamesWolf
    • 2 years ago
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    carry on

  57. estudier
    • 2 years ago
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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

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

  59. sara12345
    • 2 years ago
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    we wil get x/a = 1/4 ?

  60. estudier
    • 2 years ago
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    That was for a = 5 only....

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

  62. sara12345
    • 2 years ago
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    x/a = 2/ (sqrt(2a^2-1) + 1)

  63. estudier
    • 2 years ago
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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

  64. JamesWolf
    • 2 years ago
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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)

  65. estudier
    • 2 years ago
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    Both sides are the area of the parallelogram (the wood area) Base * "height"

  66. JamesWolf
    • 2 years ago
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    or just write on here :) |dw:1350382973068:dw|

  67. sara12345
    • 2 years ago
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    |dw:1350383010964:dw|

  68. sara12345
    • 2 years ago
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    for that ||gram, if we see bottom as base, then height = a

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

  70. sara12345
    • 2 years ago
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    correct

  71. sara12345
    • 2 years ago
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    then height =1

  72. JamesWolf
    • 2 years ago
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    okay I see thanks

  73. estudier
    • 2 years ago
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    Good geometry practice:-)

  74. sara12345
    • 2 years ago
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    as of now im not getting how to simplify quadratic formula.. il work later

  75. sara12345
    • 2 years ago
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    yesss

  76. JamesWolf
    • 2 years ago
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    back to python :)

  77. estudier
    • 2 years ago
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    Hang on I will put the quadratic.....

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

  79. sara12345
    • 2 years ago
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    x^2(1-a^2) - 2ax + 2a^2 = 0

  80. estudier
    • 2 years ago
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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...

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