He points a = (2,1,1), and b = (3,2,3) are opposite corners of a solid box whose sides are parallel with the coordinate planes. find the length of the shortest path on the surface of the box from a to b.

The shortest path can be found by drawing the net for the box, then drawing a straight line on the net between the two points. There are two choices: cross an edge of length 1, or cross an edge of length 2. The latter gives the shorter path, whose length is

... d = √(2² + (1+1)²) = 2√2 ≈ 2.828 units

_____

In the diagram, the shorter path is the dashed line. The dotted line path has length √10 ≈ 3.162 units.

___

It appears that the general solution to this sort of problem is that you want to cross the longest edge of the cuboid.

$ 0.1369 to the nearest cent ( aka hundredth decimal place) would be $ 0.14