Imagine a cake in the shape of an equilateral triangle as shown in the image bel

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   CWanamakerposted 11 years ago

   Imagine a cake in the shape of an equilateral triangle as shown in the image below. Each side of the
   
   triangle cake measures exactly 8 inches. What is the length of the shortest possible cut that will divide the cake into two pieces with equal area?
   
   

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2. 0

   calculus-geometryposted 11 years ago

   
   
   For a straight cut, I believe the shortest is parallel to one of the sides and has a length that is 1/sqrt(2) times the base length.  In this case the length is 8/sqrt(2) = 4*sqrt(2) = 5.65685. This divides the cake into an equaliateral triangle and trapezoid that have 1/2 the area of the cake. 
   
   This cut is shorter than an altitude cut, which has a length of 4*sqrt(3) =  6.9282.
   
   For a curved cut, I'm not sure. If there was a curve with a shorter length than the straight cut, it would probably be a circular arc.
   
   EDIT: Here's how to make a curved cut that has a shorter length than the straight cut. Make a 60 degree circular arc centered at one of the triangle's vertices such that the radius of the arc is 4*sqrt(3*sqrt(3)/pi).  The area of the wedge cut by the arc will be 8*sqrt(3) which is exactly half the area of the cake.  The length of the arc will be 4*sqrt(pi/sqrt(3)) = 5.38709, which is shorter than the first answer I gave.  I don't know if this is THE shortest cut.

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      tussinposted 11 years agoin reply to this

      That's cool, but why do you think the optimal solution is a circular arc?

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      CWanamakerposted 11 years agoin reply to this

      Awesome!

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      calculus-geometryposted 11 years agoin reply to this

      Circular arcs are often the solutions to minimal curve problems, so it's just a guess that this problem's solution is also a circular arc.

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      CWanamakerposted 11 years agoin reply to this

      The bisecting circular arc is the solution.

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      jpcmcposted 11 years agoin reply to this

      Darn cool answer.  After that, everyone has a headache and does not want to eat anymore.

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   tussinposted 11 years ago

   
   
   I found the longest cut..........................................

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   1. 94

      CWanamakerposted 11 years agoin reply to this

      Haha Nice!

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