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?

- anonymous

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- anonymous

|dw:1350379170293:dw|

- anonymous

Bad drawing, u probably should do your own.....:-)

- anonymous

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

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## More answers

- anonymous

|dw:1350379418357:dw|

- anonymous

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

- anonymous

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

- 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|

- anonymous

righttt sorry

- anonymous

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

- anonymous

:-)

- anonymous

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

- anonymous

|dw:1350379795901:dw|

- anonymous

asin, there is the same amount of mising area

- anonymous

|dw:1350379782354:dw|

- anonymous

like this ?

- anonymous

@sara12345 Yes, like that

- anonymous

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

- anonymous

|dw:1350379928886:dw|

- anonymous

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

- anonymous

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

- 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?

- 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.

- anonymous

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

- anonymous

that formula makes me think im wrong lol

- anonymous

whats your method estudier?

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

try with side length 5

- anonymous

I get a proportion of 1/4

- anonymous

of width of wood to area?

- anonymous

Proportion of area covered by wood

- anonymous

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

- anonymous

|dw:1350381191180:dw|

- anonymous

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

- anonymous

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

- anonymous

:)

- anonymous

|dw:1350381831150:dw|

- anonymous

A minute plz

- anonymous

K

- anonymous

i think i see what your going to do!

- anonymous

estudier that is.

- anonymous

hmm, not sure how your going to get XC

- anonymous

if thats what your going to do...

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

sorry, what is a?

- anonymous

a = A

- anonymous

a is side of square, you changed it to l

- anonymous

side of square

- anonymous

oh right sorry lol

- anonymous

carry on

- anonymous

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

- anonymous

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

- anonymous

we wil get
x/a = 1/4
?

- anonymous

That was for a = 5 only....

- anonymous

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

- anonymous

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

- anonymous

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

- 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)

- anonymous

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

- anonymous

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

- anonymous

|dw:1350383010964:dw|

- anonymous

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

- anonymous

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

- anonymous

correct

- anonymous

then height =1

- anonymous

okay I see thanks

- anonymous

Good geometry practice:-)

- anonymous

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

- anonymous

yesss

- anonymous

back to python :)

- anonymous

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

- anonymous

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

- anonymous

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

- 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...

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