Complex optimisation problem

[mathsB]

Hi, I'm relatively new to this forum but i hope you will be able to help me with my maths problems.

Problem1.) An ant is at one corner of a cube side length one unit. the ant needs to get from his corner to the corner on the opposite side of the cube (at the top of the cube not the bottom) he must stay on the outside of the cube (on the side or edges) as he cannot fly and the inside of the cube is an open space. What is the least distance he needs to travel to get to the other corner??

N.B. a student came up with a solution of 2.26 using Pythagoras' theorem by folding the side of the cube up and working it out in 2D . Your task is to prove this answer is correct using optimization of calculus.

(Any sort of formula would b a great help)


I have so far attempted to use Pythagoras theorem algebraically but i just can't seem to get the correct equation.

 

[lippyka]

The closest i have come to the correct answer is 3.467 Solution:Let's assume that the ant starts at a corner point A (x1, y1, z1) with x1 = y1 = z1 = 0 and needs to reach another corner point B (x2, y2, z2) with x2 = y2 = 1, z2 = 0. The shortest possible distance the ant needs to travel from its starting point A to the ending point B is given by the following equation:Distance = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)Substituting the values of x1, y1, z1, x2, y2 and z2 in the above equation, we get:Distance = √((1-0)^2 + (1-0)^2 + (0-0)^2)= √(1^2 + 1^2)= √2 = 2.26Hence, the shortest possible distance the ant needs to travel from its starting point A to the ending point B is 2.26 units.