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Minimization, maximization (Calculus)

An ant is walking around the outside of the cube in "straight" paths (where we define a straight path in this case as one formed by the edges of a cross section created by a plane slicing through the cube). For example, to get from point Q to point R in the picture above on the right, the ant walks along the red path. There are many different straight paths the ant can take, as you can imagine just by slicing the cube with different planes.

1. Describe and find the length of the shortest possible path the ant could take to get from point A to point G.
2. Describe and find the length of the longest path the ant could take to get from A to G.
3. Describe and find the longest possible path the ant could take to get from point A to point B.

Be sure to explain why you think your paths have the shortest and longest possible lengths.

See attached file for full problem description.

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