estudier
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?
Delete
Share
This Question is Closed
estudier
Best Response
You've already chosen the best response.
3
|dw:1350379170293:dw|
estudier
Best Response
You've already chosen the best response.
3
Bad drawing, u probably should do your own.....:-)
estudier
Best Response
You've already chosen the best response.
3
If u can see it opposite edges of the wood are aligned with corners of tile.
JamesWolf
Best Response
You've already chosen the best response.
1
|dw:1350379418357:dw|
estudier
Best Response
You've already chosen the best response.
3
@JamesWolf U should be practicing your Python!:-)
estudier
Best Response
You've already chosen the best response.
3
That pic is not correct, the edges are aligned with the corners, not in the middle...
JamesWolf
Best Response
You've already chosen the best response.
1
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|
JamesWolf
Best Response
You've already chosen the best response.
1
righttt sorry
JamesWolf
Best Response
You've already chosen the best response.
1
opposite corners, yes i should be doing python! stupid python
estudier
Best Response
You've already chosen the best response.
3
:-)
JamesWolf
Best Response
You've already chosen the best response.
1
i think |dw:1350379781386:dw|
is the same as |dw:1350379796776:dw|
estudier
Best Response
You've already chosen the best response.
3
|dw:1350379795901:dw|
JamesWolf
Best Response
You've already chosen the best response.
1
asin, there is the same amount of mising area
sara12345
Best Response
You've already chosen the best response.
0
|dw:1350379782354:dw|
sara12345
Best Response
You've already chosen the best response.
0
like this ?
estudier
Best Response
You've already chosen the best response.
3
@sara12345 Yes, like that
JamesWolf
Best Response
You've already chosen the best response.
1
|dw:1350379878261:dw|
i think the two diagrams are equivilant
sara12345
Best Response
You've already chosen the best response.
0
|dw:1350379928886:dw|
estudier
Best Response
You've already chosen the best response.
3
@JamesWolf I don't mind how u do it, your choice....
estudier
Best Response
You've already chosen the best response.
3
@sara12345 What r u going to do about the little corner bits?
JamesWolf
Best Response
You've already chosen the best response.
1
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?
estudier
Best Response
You've already chosen the best response.
3
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.
JamesWolf
Best Response
You've already chosen the best response.
1
\[A = \frac{1}{2}(l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]
JamesWolf
Best Response
You've already chosen the best response.
1
that formula makes me think im wrong lol
JamesWolf
Best Response
You've already chosen the best response.
1
whats your method estudier?
estudier
Best Response
You've already chosen the best response.
3
Let's wait, let people have a go first.....
JamesWolf
Best Response
You've already chosen the best response.
1
\[A = (\frac{1}{2}l - \frac{\sin\frac{\pi}{4}}{2})^2 - l^2\]
JamesWolf
Best Response
You've already chosen the best response.
1
alright so for a square with side length 5, i get - 14. lol
estudier
Best Response
You've already chosen the best response.
3
I haven't done it with figures, let me check....
JamesWolf
Best Response
You've already chosen the best response.
1
try with side length 5
estudier
Best Response
You've already chosen the best response.
3
I get a proportion of 1/4
JamesWolf
Best Response
You've already chosen the best response.
1
of width of wood to area?
estudier
Best Response
You've already chosen the best response.
3
Proportion of area covered by wood
JamesWolf
Best Response
You've already chosen the best response.
1
i get 1/3.888 you got exactly 1/4?
estudier
Best Response
You've already chosen the best response.
3
|dw:1350381191180:dw|
JamesWolf
Best Response
You've already chosen the best response.
1
\[2(\frac{1}{2} \times (l - \sqrt(2))^2)\] area not covered.
estudier
Best Response
You've already chosen the best response.
3
OK, unless someone comes with answer in a minute, I will post answer...
JamesWolf
Best Response
You've already chosen the best response.
1
:)
estudier
Best Response
You've already chosen the best response.
3
|dw:1350381831150:dw|
sauravshakya
Best Response
You've already chosen the best response.
0
A minute plz
estudier
Best Response
You've already chosen the best response.
3
K
JamesWolf
Best Response
You've already chosen the best response.
1
i think i see what your going to do!
JamesWolf
Best Response
You've already chosen the best response.
1
estudier that is.
JamesWolf
Best Response
You've already chosen the best response.
1
hmm, not sure how your going to get XC
JamesWolf
Best Response
You've already chosen the best response.
1
if thats what your going to do...
estudier
Best Response
You've already chosen the best response.
3
A longer way is to first see that APX and XBC are similar...
JamesWolf
Best Response
You've already chosen the best response.
1
Im not very good at geometry so im not sure how that helps
estudier
Best Response
You've already chosen the best response.
3
But a quicker way is to note that the wood area is a //gram with base x and height a
estudier
Best Response
You've already chosen the best response.
3
It is also a //gram with base XC and height 1
estudier
Best Response
You've already chosen the best response.
3
So xa = sqrt((x-a)^2 +a^2)
Square and simplify gives you a quadratic...
JamesWolf
Best Response
You've already chosen the best response.
1
sorry, what is a?
JamesWolf
Best Response
You've already chosen the best response.
1
a = A
estudier
Best Response
You've already chosen the best response.
3
a is side of square, you changed it to l
sara12345
Best Response
You've already chosen the best response.
0
side of square
JamesWolf
Best Response
You've already chosen the best response.
1
oh right sorry lol
JamesWolf
Best Response
You've already chosen the best response.
1
carry on
estudier
Best Response
You've already chosen the best response.
3
You can get the same quadratic by similar triangles as I said before:
sqrt(x^2-1)/1 = (a-x)/a
estudier
Best Response
You've already chosen the best response.
3
Solving the quadratic and calculating the proportion gives 2/ (sqrt(2a^2-1) + 1)
sara12345
Best Response
You've already chosen the best response.
0
we wil get
x/a = 1/4
?
estudier
Best Response
You've already chosen the best response.
3
That was for a = 5 only....
sara12345
Best Response
You've already chosen the best response.
0
if i go ahead and solve the quadratic,
ilg et
x/ = 2/ (sqrt(2a^2-1) + 1)
?
sara12345
Best Response
You've already chosen the best response.
0
x/a = 2/ (sqrt(2a^2-1) + 1)
estudier
Best Response
You've already chosen the best response.
3
No, you will get x = (an expression) but we want a proportion so multiply by a and divide by a^2
JamesWolf
Best Response
You've already chosen the best response.
1
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)
estudier
Best Response
You've already chosen the best response.
3
Both sides are the area of the parallelogram (the wood area)
Base * "height"
JamesWolf
Best Response
You've already chosen the best response.
1
or just write on here :) |dw:1350382973068:dw|
sara12345
Best Response
You've already chosen the best response.
0
|dw:1350383010964:dw|
sara12345
Best Response
You've already chosen the best response.
0
for that ||gram, if we see bottom as base, then height = a
JamesWolf
Best Response
You've already chosen the best response.
1
so sqrt((x-a)^2 +a^2) is working out the length from X to C
sara12345
Best Response
You've already chosen the best response.
0
correct
sara12345
Best Response
You've already chosen the best response.
0
then height =1
JamesWolf
Best Response
You've already chosen the best response.
1
okay I see thanks
estudier
Best Response
You've already chosen the best response.
3
Good geometry practice:-)
sara12345
Best Response
You've already chosen the best response.
0
as of now im not getting how to simplify quadratic formula.. il work later
sara12345
Best Response
You've already chosen the best response.
0
yesss
JamesWolf
Best Response
You've already chosen the best response.
1
back to python :)
estudier
Best Response
You've already chosen the best response.
3
Hang on I will put the quadratic.....
sara12345
Best Response
You've already chosen the best response.
0
i got :
x^2(1-a^2) + 2ax + 2a^2 = 0
sara12345
Best Response
You've already chosen the best response.
0
x^2(1-a^2) - 2ax + 2a^2 = 0
estudier
Best Response
You've already chosen the best response.
3
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...