What Is the Shortest Path an Ant Can Take Across a Cube?

[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]

Solution:Let the coordinates of the ant's starting point be (0,0,0) and the coordinates of the point he needs to reach be (1,1,1). Using the Pythagoras theorem, we can calculate the distance between the two points as:d = sqrt((1-0)^2 + (1-0)^2 + (1-0)^2) d = sqrt(3) d = 1.73To optimize the path taken by the ant, we consider the function f(x) = sqrt((1 - x)^2 + (1-x)^2 + (1-x)^2) which gives the length of the path taken by the ant as a function of the parameter x. To find the shortest path taken by the ant, we need to find the value of x which minimizes the function f(x). We do this by taking the first derivative of the function and setting it equal to 0. This gives us the equation: f'(x) = 0 2(1-x) = 0 x = 1 Substituting this value of x into our function f(x), we find that the shortest path taken by the ant is: d = sqrt((1-1)^2 + (1-1)^2 + (1-1)^2) d = sqrt(3) d = 1.73 Thus, the ant needs to travel a distance of 1.73 units to reach the opposite corner of the cube. This answer is the same as the one obtained using Pythagoras theorem, thus showing that the answer of 2.26 obtained by folding the side of the cube up is incorrect.