A farmer wishes to enclose the maximum possible area with 100 meters of fence. The pasture is bordered by a straight cliff, which may be used as part of the fence. What is the maximum area that can be enclosed?

The actual fencing used should be in the form of a circular arc, as the local topology at any point is the same as on a full circle, which would be the most efficient enclosure if a cliff were not available.

Then the question becomes What fraction of a full circle should the arc be?  Consider the topology around the points where the circular arc meets the cliff. If a small increment of fence is perpendicular to the cliff, it has pushed the remainder of the fence as far as possible away from the cliff, expanding the area to its maximum. Any deviation from perpendicular will require pulling in the  remainder of the fence, with only a small return if it is made outward.

So the fence should be made into a semicircle. The radius of the semicircle would then be 100/pi meters, and the area would be pi*(100/pi)^2 / 2 = 5000/pi = 1591.5... square meters.

Posted by Charlie on 2004-06-18 10:08:07