I don't really understand the author's comment that "it cannot be solved without very sophisticated math, rarely studied even in college". That would be true if

I don't really understand the author's comment that "it cannot be solved without very sophisticated math, rarely studied even in college". That would be true if the lengths of the fences were arbitrary or the fences were potentially curved, but it's relatively straightforward to develop a proof of max area given that all the fences are the same length. Step 1 would be to demonstrate that the shape with max area has to be convex. This is pretty straightforward, and a pretty decent example for advanced middle school kids. Once you've got convexity proven, then the total area covered becomes the sum of triangular wedges. There are a variety of ways here to arrive at the maximum, either by a sequence of replacements (probably most appropriate for advanced middle schoolers) or a formulation in terms of Langrange multipliers (for advanced high school students). I think a lot of students would find the proof pretty cool.