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xapproachesinfinity

  • one year ago

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

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  1. xapproachesinfinity
    • one year ago
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    area not volume

  2. ganeshie8
    • one year ago
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    wouldn't the area increase by the same factor ?

  3. anonymous
    • one year ago
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    suppose it is a line?

  4. ganeshie8
    • one year ago
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    question would be invalid then, area for a line makes no sense

  5. xapproachesinfinity
    • one year ago
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    that's why i said determine what object

  6. anonymous
    • one year ago
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    suppose it is a circle?

  7. acxbox22
    • one year ago
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    in the first dimension it could

  8. ganeshie8
    • one year ago
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    circle becomes ellipse and increasing the major axis by "k" times increases the area by "k" too so we're fine

  9. xapproachesinfinity
    • one year ago
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    well that's why i asked we cannot generalize to any object

  10. ganeshie8
    • one year ago
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    the way i see it, we can generalize... unless you have a counter example

  11. acxbox22
    • one year ago
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    what if it is an irregular polygon?

  12. anonymous
    • one year ago
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    i guess i am confused as to what the "length" of a shape it, but maybe that is just my confusion

  13. ganeshie8
    • one year ago
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    it doesn't matter how the shape looks

  14. acxbox22
    • one year ago
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    well like sat said...it depends on what length is for each shape...

  15. xapproachesinfinity
    • one year ago
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    i said any object and only scaling the length

  16. xapproachesinfinity
    • one year ago
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    well for regular gons scaling one side with some factor scales the area with the same factor

  17. xapproachesinfinity
    • one year ago
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    i mean not gons just rectangle, square....

  18. ganeshie8
    • one year ago
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    by length your friend means that he is scaling in "one" dimension

  19. xapproachesinfinity
    • one year ago
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    yes one dimension scaling

  20. ganeshie8
    • one year ago
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    imagine the object is in a paper and you "stretch" that paper horizontally by a factor of "k"

  21. ganeshie8
    • one year ago
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    the area scales by a factor of "k" too, no matter how the shape looks

  22. acxbox22
    • one year ago
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    why did u friend say that you know know nothing about math if you are right then? :P

  23. xapproachesinfinity
    • one year ago
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    hmm seems that way! how can we justify the general idea

  24. xapproachesinfinity
    • one year ago
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    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

  25. xapproachesinfinity
    • one year ago
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    but he refused and said the object matters not heheh

  26. xapproachesinfinity
    • one year ago
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    feel dumb now lol

  27. anonymous
    • one year ago
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    I am dumb

  28. xapproachesinfinity
    • one year ago
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    Noway! you are a mathematician, can't be dumb haha

  29. xapproachesinfinity
    • one year ago
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    at any rate, i think my friend and i had ego clash

  30. xapproachesinfinity
    • one year ago
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    @ganeshie8 can't see how we can generalize with any object? first we need to pin point what does length mean for any object

  31. ganeshie8
    • one year ago
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    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|

  32. ganeshie8
    • one year ago
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    The claim is that the red closed shape takes twice the area of black shape

  33. xapproachesinfinity
    • one year ago
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    hmm i'm not really convinced there is a subtlety to how do you know it is scaled by the same factor

  34. xapproachesinfinity
    • one year ago
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    looks like a topology problem no?

  35. ganeshie8
    • one year ago
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    familiar with jacobians ?

  36. ganeshie8
    • one year ago
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    it is a simple change of variables problem : \[X = kx\\~\\Y=y\] find the jacobian

  37. xapproachesinfinity
    • one year ago
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    well the word is familiar to me but i don't remember how to do it anymore haha

  38. ganeshie8
    • one year ago
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    jacobian gives you the scale factor for areas between the two coordinate systems which is exactly what we need

  39. xapproachesinfinity
    • one year ago
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    hmm i see

  40. ganeshie8
    • one year ago
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    \[J =\begin{vmatrix} X_x&X_y\\Y_x&Y_y\end{vmatrix} = \begin{vmatrix}k&0\\0&1\end{vmatrix} = k\] therefore \[dXdY = k\,dxdy\] area scales by a factor of "k" when you scale the shape in one dimension by a factor of k

  41. xapproachesinfinity
    • one year ago
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    Fair enough

  42. xapproachesinfinity
    • one year ago
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    thanks a lot!

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