xapproachesinfinity
  • xapproachesinfinity
guys! here is something. my friend asked me a question if we increased the length of an object by x how much its area has increased. so my answer was what kind of object it is. we need to know what kind of object we are dealing to know how much it is been scaled but my dear friend said no you know nothing about math lol so what do you guys say
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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xapproachesinfinity
  • xapproachesinfinity
area not volume
ganeshie8
  • ganeshie8
wouldn't the area increase by the same factor ?
anonymous
  • anonymous
suppose it is a line?

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ganeshie8
  • ganeshie8
question would be invalid then, area for a line makes no sense
xapproachesinfinity
  • xapproachesinfinity
that's why i said determine what object
anonymous
  • anonymous
suppose it is a circle?
acxbox22
  • acxbox22
in the first dimension it could
ganeshie8
  • ganeshie8
circle becomes ellipse and increasing the major axis by "k" times increases the area by "k" too so we're fine
xapproachesinfinity
  • xapproachesinfinity
well that's why i asked we cannot generalize to any object
ganeshie8
  • ganeshie8
the way i see it, we can generalize... unless you have a counter example
acxbox22
  • acxbox22
what if it is an irregular polygon?
anonymous
  • anonymous
i guess i am confused as to what the "length" of a shape it, but maybe that is just my confusion
ganeshie8
  • ganeshie8
it doesn't matter how the shape looks
acxbox22
  • acxbox22
well like sat said...it depends on what length is for each shape...
xapproachesinfinity
  • xapproachesinfinity
i said any object and only scaling the length
xapproachesinfinity
  • xapproachesinfinity
well for regular gons scaling one side with some factor scales the area with the same factor
xapproachesinfinity
  • xapproachesinfinity
i mean not gons just rectangle, square....
ganeshie8
  • ganeshie8
by length your friend means that he is scaling in "one" dimension
xapproachesinfinity
  • xapproachesinfinity
yes one dimension scaling
ganeshie8
  • ganeshie8
imagine the object is in a paper and you "stretch" that paper horizontally by a factor of "k"
ganeshie8
  • ganeshie8
the area scales by a factor of "k" too, no matter how the shape looks
acxbox22
  • acxbox22
why did u friend say that you know know nothing about math if you are right then? :P
xapproachesinfinity
  • xapproachesinfinity
hmm seems that way! how can we justify the general idea
xapproachesinfinity
  • xapproachesinfinity
well i didn't really give him an answer, i just said what kind of object are you trying to scale so i know how to answer you
xapproachesinfinity
  • xapproachesinfinity
but he refused and said the object matters not heheh
xapproachesinfinity
  • xapproachesinfinity
feel dumb now lol
anonymous
  • anonymous
I am dumb
xapproachesinfinity
  • xapproachesinfinity
Noway! you are a mathematician, can't be dumb haha
xapproachesinfinity
  • xapproachesinfinity
at any rate, i think my friend and i had ego clash
xapproachesinfinity
  • xapproachesinfinity
@ganeshie8 can't see how we can generalize with any object? first we need to pin point what does length mean for any object
ganeshie8
  • ganeshie8
This is not a proof, just trying to convince myself more consider below shape |dw:1432872302786:dw| stretch it horizontally by a factor of 2 |dw:1432872323889:dw|
ganeshie8
  • ganeshie8
The claim is that the red closed shape takes twice the area of black shape
xapproachesinfinity
  • xapproachesinfinity
hmm i'm not really convinced there is a subtlety to how do you know it is scaled by the same factor
xapproachesinfinity
  • xapproachesinfinity
looks like a topology problem no?
ganeshie8
  • ganeshie8
familiar with jacobians ?
ganeshie8
  • ganeshie8
it is a simple change of variables problem : \[X = kx\\~\\Y=y\] find the jacobian
xapproachesinfinity
  • xapproachesinfinity
well the word is familiar to me but i don't remember how to do it anymore haha
ganeshie8
  • ganeshie8
jacobian gives you the scale factor for areas between the two coordinate systems which is exactly what we need
xapproachesinfinity
  • xapproachesinfinity
hmm i see
ganeshie8
  • ganeshie8
\[J =\begin{vmatrix} X_x&X_y\\Y_x&Y_y\end{vmatrix} = \begin{vmatrix}k&0\\0&1\end{vmatrix} = k\] therefore \[dXdY = k\,dxdy\] area scales by a factor of "k" when you scale the shape in one dimension by a factor of k
xapproachesinfinity
  • xapproachesinfinity
Fair enough
xapproachesinfinity
  • xapproachesinfinity
thanks a lot!

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