Question

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

Answer 1

area not volume

Answer 2

wouldn't the area increase by the same factor ?

Answer 3

suppose it is a line?

Answer 4

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

Answer 5

that's why i said determine what object

Answer 6

suppose it is a circle?

Answer 7

in the first dimension it could

Answer 8

circle becomes ellipse
and increasing the major axis by "k" times increases the area by "k" too

so we're fine

Answer 9

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

Answer 10

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

Answer 11

what if it is an irregular polygon?

Answer 12

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

Answer 13

it doesn't matter how the shape looks

Answer 14

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

Answer 15

i said any object
and only scaling the length

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

yes one dimension scaling

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

but he refused and said the object matters not heheh

Answer 26

feel dumb now lol

Answer 27

I am dumb

Answer 28

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

Answer 29

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

Answer 30

@ganeshie8 can't see how we can generalize with any object?
first we need to pin point what does length mean for any object

Answer 31

This is not a proof, just trying to convince myself more
consider below shape



stretch it horizontally by a factor of 2

Answer 32

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

Answer 33

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

Answer 34

looks like a topology problem no?

Answer 35

familiar with jacobians ?

Answer 36

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


find the jacobian

Answer 37

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

Answer 38

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

Answer 39

hmm i see

Answer 40

\[J =\begin{vmatrix} X_x&X_y\\Y_x&Y_y\end{vmatrix} = \begin{vmatrix}k&0\\0&1\end{vmatrix} = k\]

therefore \[dXdY = k\,dxdy\]

area scales by a factor of "k" when you scale the shape in one dimension by a factor of k

Answer 41

Fair enough

Answer 42

thanks a lot!