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?

1. estudier

|dw:1350379170293:dw|

2. estudier

3. estudier

If u can see it opposite edges of the wood are aligned with corners of tile.

4. JamesWolf

|dw:1350379418357:dw|

5. estudier

@JamesWolf U should be practicing your Python!:-)

6. estudier

That pic is not correct, the edges are aligned with the corners, not in the middle...

7. JamesWolf

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

righttt sorry

9. JamesWolf

opposite corners, yes i should be doing python! stupid python

10. estudier

:-)

11. JamesWolf

i think |dw:1350379781386:dw| is the same as |dw:1350379796776:dw|

12. estudier

|dw:1350379795901:dw|

13. JamesWolf

asin, there is the same amount of mising area

14. sara12345

|dw:1350379782354:dw|

15. sara12345

like this ?

16. estudier

@sara12345 Yes, like that

17. JamesWolf

|dw:1350379878261:dw| i think the two diagrams are equivilant

18. sara12345

|dw:1350379928886:dw|

19. estudier

@JamesWolf I don't mind how u do it, your choice....

20. estudier

@sara12345 What r u going to do about the little corner bits?

21. JamesWolf

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

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

$A = \frac{1}{2}(l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2$

24. JamesWolf

that formula makes me think im wrong lol

25. JamesWolf

26. estudier

Let's wait, let people have a go first.....

27. JamesWolf

$A = (\frac{1}{2}l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2$

28. JamesWolf

alright so for a square with side length 5, i get - 14. lol

29. estudier

I haven't done it with figures, let me check....

30. JamesWolf

try with side length 5

31. estudier

I get a proportion of 1/4

32. JamesWolf

of width of wood to area?

33. estudier

Proportion of area covered by wood

34. JamesWolf

i get 1/3.888 you got exactly 1/4?

35. estudier

|dw:1350381191180:dw|

36. JamesWolf

$2(\frac{1}{2} \times (l - \sqrt(2))^2)$ area not covered.

37. estudier

OK, unless someone comes with answer in a minute, I will post answer...

38. JamesWolf

:)

39. estudier

|dw:1350381831150:dw|

40. sauravshakya

A minute plz

41. estudier

K

42. JamesWolf

i think i see what your going to do!

43. JamesWolf

estudier that is.

44. JamesWolf

hmm, not sure how your going to get XC

45. JamesWolf

if thats what your going to do...

46. estudier

A longer way is to first see that APX and XBC are similar...

47. JamesWolf

Im not very good at geometry so im not sure how that helps

48. estudier

But a quicker way is to note that the wood area is a //gram with base x and height a

49. estudier

It is also a //gram with base XC and height 1

50. estudier

So xa = sqrt((x-a)^2 +a^2) Square and simplify gives you a quadratic...

51. JamesWolf

sorry, what is a?

52. JamesWolf

a = A

53. estudier

a is side of square, you changed it to l

54. sara12345

side of square

55. JamesWolf

oh right sorry lol

56. JamesWolf

carry on

57. estudier

You can get the same quadratic by similar triangles as I said before: sqrt(x^2-1)/1 = (a-x)/a

58. estudier

Solving the quadratic and calculating the proportion gives 2/ (sqrt(2a^2-1) + 1)

59. sara12345

we wil get x/a = 1/4 ?

60. estudier

That was for a = 5 only....

61. sara12345

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

62. sara12345

x/a = 2/ (sqrt(2a^2-1) + 1)

63. estudier

No, you will get x = (an expression) but we want a proportion so multiply by a and divide by a^2

64. JamesWolf

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

Both sides are the area of the parallelogram (the wood area) Base * "height"

66. JamesWolf

or just write on here :) |dw:1350382973068:dw|

67. sara12345

|dw:1350383010964:dw|

68. sara12345

for that ||gram, if we see bottom as base, then height = a

69. JamesWolf

so sqrt((x-a)^2 +a^2) is working out the length from X to C

70. sara12345

correct

71. sara12345

then height =1

72. JamesWolf

okay I see thanks

73. estudier

Good geometry practice:-)

74. sara12345

as of now im not getting how to simplify quadratic formula.. il work later

75. sara12345

yesss

76. JamesWolf

back to python :)

77. estudier

Hang on I will put the quadratic.....

78. sara12345

i got : x^2(1-a^2) + 2ax + 2a^2 = 0

79. sara12345

x^2(1-a^2) - 2ax + 2a^2 = 0

80. estudier

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...