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The greatest area of a shape with a given perimeter is achieved when that shape is a circle. Rigorous proof of this is beyond the scope of the given problem, but once this is assumed as true, the calculations become easy:
If the length of a string is 132cm, and it is layed out in a circle, its length will be the length of the circle's circumferrence. Thus
2(pi)r = 132
r = 132 / 2(pi)
The area of this circle is
(pi)r^2 =
(pi)(132/ 2(pi))^2 =
132^2 / 4(pi) =
1386.56
The minimum area is of course achieved by folding the string in two and is equal to zero. |
