## anonymous 3 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. anonymous

|dw:1350379170293:dw|

2. anonymous

3. anonymous

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

4. anonymous

|dw:1350379418357:dw|

5. anonymous

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

6. anonymous

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

7. anonymous

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. anonymous

righttt sorry

9. anonymous

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

10. anonymous

:-)

11. anonymous

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

12. anonymous

|dw:1350379795901:dw|

13. anonymous

asin, there is the same amount of mising area

14. anonymous

|dw:1350379782354:dw|

15. anonymous

like this ?

16. anonymous

@sara12345 Yes, like that

17. anonymous

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

18. anonymous

|dw:1350379928886:dw|

19. anonymous

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

20. anonymous

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

21. anonymous

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. anonymous

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. anonymous

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

24. anonymous

that formula makes me think im wrong lol

25. anonymous

26. anonymous

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

27. anonymous

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

28. anonymous

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

29. anonymous

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

30. anonymous

try with side length 5

31. anonymous

I get a proportion of 1/4

32. anonymous

of width of wood to area?

33. anonymous

Proportion of area covered by wood

34. anonymous

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

35. anonymous

|dw:1350381191180:dw|

36. anonymous

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

37. anonymous

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

38. anonymous

:)

39. anonymous

|dw:1350381831150:dw|

40. anonymous

A minute plz

41. anonymous

K

42. anonymous

i think i see what your going to do!

43. anonymous

estudier that is.

44. anonymous

hmm, not sure how your going to get XC

45. anonymous

if thats what your going to do...

46. anonymous

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

47. anonymous

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

48. anonymous

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

49. anonymous

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

50. anonymous

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

51. anonymous

sorry, what is a?

52. anonymous

a = A

53. anonymous

a is side of square, you changed it to l

54. anonymous

side of square

55. anonymous

oh right sorry lol

56. anonymous

carry on

57. anonymous

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

58. anonymous

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

59. anonymous

we wil get x/a = 1/4 ?

60. anonymous

That was for a = 5 only....

61. anonymous

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

62. anonymous

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

63. anonymous

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

64. anonymous

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. anonymous

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

66. anonymous

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

67. anonymous

|dw:1350383010964:dw|

68. anonymous

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

69. anonymous

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

70. anonymous

correct

71. anonymous

then height =1

72. anonymous

okay I see thanks

73. anonymous

Good geometry practice:-)

74. anonymous

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

75. anonymous

yesss

76. anonymous

back to python :)

77. anonymous

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

78. anonymous

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

79. anonymous

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

80. anonymous

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...