A hiker is returning to a campsite that is 2 miles east and 5 miles north from their current position. The hiker is currently walking east along a path at a pace of 5 miles per hour. The hiker plans to leave the path and walk in a straight line, diagonally towards the cabin through the woods. The hiker can only walk at a pace of 2 miles per hour through the woods. Where should the hiker leave the path and turn into the woods if they want to get to the campsite as quickly as possible?

Please Include a well-labeled figure, clearly defined objective function of one variable, interval for which that variable is logical, and supporting mathematical statements to justify that the optimal result is achieved for the function on the interval within a grammatically correct exposition.