posted by Regina .
1. A gardener has 140 feet of fencing to fence in a rectangular vegetable garden. Find the dimensions of the largest area he can fence. Find the possible rectangular area he can enclose.
2. Suppose a farmer has a large piece of land and he wants to make a rectangular fence for his animals, but he has no money to buy more wood for the fence. The total length of the fence is fixed to 250 meters. What should be the width and length of the rectangle such that the area is maximized?
It can be shown by calculus that the maximum area of a polygon enclosed by a fixed length of perimeter is a regular polygon, i.e. a polygon with all sides and interior angles equal.
It follows from the above that the maximum area enclosed by a fixed perimeter is a circle, equivalent to a polygon with "infinite" number of sides.