- 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

- schrodinger

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- xapproachesinfinity

area not volume

- ganeshie8

wouldn't the area increase by the same factor ?

- anonymous

suppose it is a line?

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## More answers

- ganeshie8

question would be invalid then, area for a line makes no sense

- xapproachesinfinity

that's why i said determine what object

- anonymous

suppose it is a circle?

- acxbox22

in the first dimension it could

- ganeshie8

circle becomes ellipse
and increasing the major axis by "k" times increases the area by "k" too
so we're fine

- xapproachesinfinity

well that's why i asked
we cannot generalize to any object

- ganeshie8

the way i see it, we can generalize... unless you have a counter example

- acxbox22

what if it is an irregular polygon?

- anonymous

i guess i am confused as to what the "length" of a shape it, but maybe that is just my confusion

- ganeshie8

it doesn't matter how the shape looks

- acxbox22

well like sat said...it depends on what length is for each shape...

- xapproachesinfinity

i said any object
and only scaling the length

- xapproachesinfinity

well for regular gons
scaling one side with some factor scales the area with the same factor

- xapproachesinfinity

i mean not gons
just rectangle, square....

- ganeshie8

by length your friend means that he is scaling in "one" dimension

- xapproachesinfinity

yes one dimension scaling

- ganeshie8

imagine the object is in a paper
and you "stretch" that paper horizontally by a factor of "k"

- ganeshie8

the area scales by a factor of "k" too, no matter how the shape looks

- acxbox22

why did u friend say that you know know nothing about math if you are right then? :P

- xapproachesinfinity

hmm seems that way! how can we justify the general idea

- 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

but he refused and said the object matters not heheh

- xapproachesinfinity

feel dumb now lol

- anonymous

I am dumb

- xapproachesinfinity

Noway! you are a mathematician, can't be dumb haha

- xapproachesinfinity

at any rate, i think my friend and i had ego clash

- 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

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

The claim is that the red closed shape takes twice the area of black shape

- xapproachesinfinity

hmm i'm not really convinced
there is a subtlety to how do you know it is scaled by the same factor

- xapproachesinfinity

looks like a topology problem no?

- ganeshie8

familiar with jacobians ?

- ganeshie8

it is a simple change of variables problem :
\[X = kx\\~\\Y=y\]
find the jacobian

- xapproachesinfinity

well the word is familiar to me but i don't remember how to do it anymore haha

- ganeshie8

jacobian gives you the scale factor for areas between the two coordinate systems
which is exactly what we need

- xapproachesinfinity

hmm i see

- 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

Fair enough

- xapproachesinfinity

thanks a lot!

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