This is the kind of thing that comes up in story problems in Calculus often. If you can travel in/over one medium at one speed, and in/over another medium at a different speed, what is the optimum path to minimize your travel time.
An example of this problem would be if there is a drowning swimmer 100 meters offshore, you are 300 meters from the point on the shoreline closest to the swimmer, and you can run at 15mph and swim at 2mph, how far do you run along the shoreline before going into the water to get to the swimmer as quickly as possible?
The fact that Randall shows two different paths over the "grass" makes me think that he was thinking more along the line of obsessively optimizing his path rather than about whether it might be acceptable or not to walk over the grass. -- mwburden 22.214.171.124 21:23, 13 December 2012 (UTC)
Along similar lines, this mathematician's dog uses Calculus (albeit at an intuitive, rather than mathematical level) to optimize the path that it takes to retrieve the ball from the water. -- mwburden 126.96.36.199 21:27, 13 December 2012 (UTC)
This particular situation is less interesting, since the walker's speed is the same for all three paths! This is seen by the times being directly proportional to the distances. Normally, the off-normal-path is at a lower speed, but some shorter path still gives the smallest time.DrMath 08:22, 14 October 2013 (UTC)