0
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Given a box with its 8 corners at

(0,0,0)
(481,0,0)
(0,53,0)
(0,0,490)
(481,53,0)
(481,0,490)
(0,53,490)
(481,53,490)

An ant is positioned at (0,0,0) and would like to head to the point (481,51,256). The ant can only walk at the surface of the box. What is the shortest path of the ant towards the point?

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    $\begingroup$ Unfold the box and have the ant walk a straight line. There are a few ways to unfold it, you have to try them. $\endgroup$ Mar 5 '20 at 6:03
  • $\begingroup$ I tried the two possible ways to unfold, which yielded me 590.39 and 738.76.But the answer from my teacher was 573.38 $\endgroup$ Mar 5 '20 at 6:08
  • $\begingroup$ @RossMillikan. Your nice solution remembered me the Gordian knot. Cheers :-) $\endgroup$ Mar 5 '20 at 9:57
  • $\begingroup$ Isn't the answer $573.3\color{red}28$ ? $\endgroup$
    – user65203
    Mar 5 '20 at 16:00
  • $\begingroup$ Technically my teacher told me to write the answer as the distance squared, so the answer is 328770, after taking the square root, I got 573.3847 $\endgroup$ Mar 6 '20 at 6:15
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There are 3 combinations to unfold In each combination you take 2 coordinates and 3rd coordinate to be perpendicular

C-1 : dist^2 = (x+y)^2 + z^2....dist = 590.39

C-2 : dist^2 = (x+z)^2 + y^2....dist = 738.36

C-3 : dist^2 = (y+z)^2 + x^2....dist - 570.62

So min distance is with C-3 which is 570.62

You are unfolding the planes - 1 xz plane at y=0 and 2 xy plane at z=256

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You only need to unfold the box and draw it's net, the rest is just Pythagoras. enter image description here

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10
  • $\begingroup$ The target endpoint is not a corner. $\endgroup$
    – user65203
    Mar 5 '20 at 15:04
  • $\begingroup$ @YvesDaoust, Thank you, it is fixed now. $\endgroup$
    – Seyed
    Mar 5 '20 at 15:55
  • $\begingroup$ This still doesn't match the "official" answer, which seems to be $\sqrt{481^2+(256+56)^2}$. $\endgroup$
    – user65203
    Mar 5 '20 at 15:59
  • $\begingroup$ @YvesDaoust, Apparently the official answer is not correct. $\endgroup$
    – Seyed
    Mar 5 '20 at 16:01
  • $\begingroup$ I got the coordinates wrong. The height of the box should be 490. I'm terribly sorry $\endgroup$ Mar 6 '20 at 6:25

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