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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?

Mathematics
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|dw:1350379170293:dw|
Bad drawing, u probably should do your own.....:-)
If u can see it opposite edges of the wood are aligned with corners of tile.

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Other answers:

|dw:1350379418357:dw|
@JamesWolf U should be practicing your Python!:-)
That pic is not correct, the edges are aligned with the corners, not in the middle...
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|
righttt sorry
opposite corners, yes i should be doing python! stupid python
:-)
i think |dw:1350379781386:dw| is the same as |dw:1350379796776:dw|
|dw:1350379795901:dw|
asin, there is the same amount of mising area
|dw:1350379782354:dw|
like this ?
@sara12345 Yes, like that
|dw:1350379878261:dw| i think the two diagrams are equivilant
|dw:1350379928886:dw|
@JamesWolf I don't mind how u do it, your choice....
@sara12345 What r u going to do about the little corner bits?
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?
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.
\[A = \frac{1}{2}(l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]
that formula makes me think im wrong lol
whats your method estudier?
Let's wait, let people have a go first.....
\[A = (\frac{1}{2}l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]
alright so for a square with side length 5, i get - 14. lol
I haven't done it with figures, let me check....
try with side length 5
I get a proportion of 1/4
of width of wood to area?
Proportion of area covered by wood
i get 1/3.888 you got exactly 1/4?
|dw:1350381191180:dw|
\[2(\frac{1}{2} \times (l - \sqrt(2))^2)\] area not covered.
OK, unless someone comes with answer in a minute, I will post answer...
:)
|dw:1350381831150:dw|
A minute plz
K
i think i see what your going to do!
estudier that is.
hmm, not sure how your going to get XC
if thats what your going to do...
A longer way is to first see that APX and XBC are similar...
Im not very good at geometry so im not sure how that helps
But a quicker way is to note that the wood area is a //gram with base x and height a
It is also a //gram with base XC and height 1
So xa = sqrt((x-a)^2 +a^2) Square and simplify gives you a quadratic...
sorry, what is a?
a = A
a is side of square, you changed it to l
side of square
oh right sorry lol
carry on
You can get the same quadratic by similar triangles as I said before: sqrt(x^2-1)/1 = (a-x)/a
Solving the quadratic and calculating the proportion gives 2/ (sqrt(2a^2-1) + 1)
we wil get x/a = 1/4 ?
That was for a = 5 only....
if i go ahead and solve the quadratic, ilg et x/ = 2/ (sqrt(2a^2-1) + 1) ?
x/a = 2/ (sqrt(2a^2-1) + 1)
No, you will get x = (an expression) but we want a proportion so multiply by a and divide by a^2
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)
Both sides are the area of the parallelogram (the wood area) Base * "height"
or just write on here :) |dw:1350382973068:dw|
|dw:1350383010964:dw|
for that ||gram, if we see bottom as base, then height = a
so sqrt((x-a)^2 +a^2) is working out the length from X to C
correct
then height =1
okay I see thanks
Good geometry practice:-)
as of now im not getting how to simplify quadratic formula.. il work later
yesss
back to python :)
Hang on I will put the quadratic.....
i got : x^2(1-a^2) + 2ax + 2a^2 = 0
x^2(1-a^2) - 2ax + 2a^2 = 0
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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