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What's the shortest length here?

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  1. Jun 26, 2017 #1
    1. The problem statement, all variables and given/known data

    Question: An ant starts at one vertex of a solid cube with side of unity length. Calculate
    the distance of the shortest route the ant can take to the furthest vertex
    from the starting point.
    Now, in the answer, the cuboid is unfolded and a rectangle of side 1 and 2 is formed. Calculating the hyptoneuse gives the distance.

    That was the proposed solution. I wanted to calculate it via sqrt(1+1+1). Why is my answer wrong?


    2. Relevant equations
    H^2=P^2+B^2

    3. The attempt at a solution
    Sqrt( 1^2+1^2+1^2)
     
  2. jcsd
  3. Jun 26, 2017 #2

    ehild

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    Yes, the shortest distance between the furthest vertices is along the body diagonal (the red line in the picture) but can the ant move along it?
    upload_2017-6-26_7-35-0.png
     
  4. Jun 26, 2017 #3
    Then it should be 1+sqrt2
     
  5. Jun 26, 2017 #4
    And the answer is sqrt5
     
  6. Jun 26, 2017 #5

    ehild

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    Why? It is not the shortest distance.
    Follow the hint in the solution, unfold the cube. The shortest way between the red points is the straight line segment connecting them.
    upload_2017-6-26_8-44-22.png
     
  7. Jun 26, 2017 #6
    But how can an ant travel along the diagonal you've made?
     
  8. Jun 26, 2017 #7

    ehild

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    The ant can climb on a sheet of paper, either it is folded or unfolded.
    upload_2017-6-26_10-38-37.png

    Cut the pattern in Post #5 and fold it to make a cube.
     
  9. Jun 26, 2017 #8
    Got it. But without unfolding it, how'd you get the intuition, or prove rationally, that it is shortest path?
     
  10. Jun 26, 2017 #9

    ehild

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    Unfolded, it is the same sheet of paper, as the cube was made of. On the plane sheet, the shortest path between two points is the straight line connecting them. So unfold the cube, connect the two points with a straight line, fold back making the cube, and you see the shortest path on the surface of the cube.
    This method is very easy and can be used for other shapes which can be unfolded into a plane pattern.
    But you can do it mathematically. The path should cross an edge, see figure. Write up the length of the path as the sum s1+s2, in terms of x. Find the minimum.
    upload_2017-6-26_12-31-27.png
     
    Last edited: Jun 26, 2017
  11. Jun 26, 2017 #10

    Ray Vickson

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    Sometimes problems are difficult to solve one way, but quite easy if looked at in another way. This is one of those problems: the "unfolding" method makes it straightforward.

    The alternative would be to not unfold the cube, but to express the distance in terms of some relevant variable or variables, then perform a minimization, using calculus, for example.
     
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