The ratio of the area of a polygon to the perimeter of the polygon increases as the number of sides increases and for two polygons with the same number of sides it is higher if the length of the sides is the same. A circle can be considered to be a polygon made up of an infinite number of equal sides. In the given problem the island has a shore line of 100 miles. Its area would be maximum if the island is circular in shape. If the radius of the circle is r, the perimeter `2*pi*r = 100`

=> `r = 100/(2*pi)`

The area of the circle would be `pi*r^2` = `pi*(100/(2*pi))^2` = `(10000*pi)/(4*pi^2) = 2500/pi`

The maximum area of the island can be `2500/pi ` square miles.