ganeshie8
  • ganeshie8
Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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ganeshie8
  • ganeshie8
|dw:1435819109185:dw|
beginnersmind
  • beginnersmind
This must be fairly easy with variational methods but I'm wondering if there's a more elementary method that only uses calculus. I think I can sort of see that at every point the curve must be perpendicular to the line segment connecting it to the vertex, but I wouldn't know how to prove it.|dw:1435826489623:dw| The image sort of describes my intuition. Since over the small distance r doesn't change, the biggest area we can get is by going perpendicular to r. I guess this would imply that we need to go along a circle centered on the facing vertex.
beginnersmind
  • beginnersmind
"the biggest area we can get is by going perpendicular to r." I meant : the largest amount of area we can get for a given additional length ds.

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ikram002p
  • ikram002p
so i cant make only lne ?
ganeshie8
  • ganeshie8
You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter.
ikram002p
  • ikram002p
what type of circle ?
ganeshie8
  • ganeshie8
@ikram002p median of a triangle does divide the triangle into two parts of equal area, but it need not be of minimum length right
ikram002p
  • ikram002p
|dw:1435827736653:dw|
ikram002p
  • ikram002p
or |dw:1435827776826:dw|
ganeshie8
  • ganeshie8
yes i think beginnersmind is referring to the second one
ikram002p
  • ikram002p
well we always can construct a short cut for curves |dw:1435827828432:dw|
beginnersmind
  • beginnersmind
Yeah, that one. I get 0.6734 for length, while using a line takes 0.707 (1 over squareroot 2)
beginnersmind
  • beginnersmind
"You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter." Thanks. I think I got it now. Let's assume there's another curve with a shorter length that encloses the same amount of area. Then we could copy it 5 more times to create a closed curve which has the same area as a circle but smaller perimeter, which would be a contradiction.|dw:1435828385612:dw|
ikram002p
  • ikram002p
If u wanna the smallest closed curve inside a triangle that touches its edges that would be a circle.
ganeshie8
  • ganeshie8
Excellent! that gives us a closed curve inside a hexagon. The closed curve divides the hexagon into two parts of equal area
ikram002p
  • ikram002p
Lovely <3
beginnersmind
  • beginnersmind
Really cool puzzle. Seems almost impossible at first but actually all you need is a good idea and the fact about the circle having the smallest perimeter for a given enclosed area.
ganeshie8
  • ganeshie8
|dw:1435915373490:dw|

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