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A calculus class once took a field trip into the countryside to demonstrate the power of mathematics. The class strolled out into a grassy meadow where they found a length of fencing rolled up and leaning against the side of a barn. The challenge was to use the fencing to build the largest corral for some livestock.
The first suggestion from the students was to use the fence to bound a square area. This was a good suggestion because of all rectangular shapes with fixed perimeter, the square has the most area. So the students dutifully began set up a square area, each side measuring a quarter of the total length of the availible fencing.
Another enterprising student remarked that if they didn't restrict themselves to rectangles, they could do much better with a circular shaped region. Groaning in the realization that this meant more physical labor, the students began to devise a way to construct a circular corral.
A flash of inspiration then struck yet another student. If the fencing was formed into a rectangle with one of the sides being the barn, they might be able to do even better than the circle. More importantly, they could go back to constructing a nice rectangular region. After the corral was constructed against the barn the professor made a few measurements from within the corral to verify that this was, in fact, a better solution. Shaking his head, he made his way over to the corner where his students were now relaxing against the barn. He said that they could do so much better. Winded from all the work, the students refused to move any more fencing. If it was going to change, they said, the professor would be the one doing all the work.
Happily, the professor walked over to the loose corner of the fencing. He took one end of the fencing in hand and wrapped himself as tightly as possible and then beaming with pride declared that he had found the optimal solution. The class scoffed.
"But you can barely fit yourself in there!" protested one student.
"Untrue! I'm standing on the outside."