Rectangle with a perimeter of 18 inches . . .
-- The greatest possible area is achieved when you use your perimeter to
enclose a circle.
An 18-inch string formed into a circle encloses about 25.78 square inches.
-- Since we're told to do it in a rectangle, the next greatest possible area is
when you use your perimeter to enclose a square. An 18-inch string formed
into a square with four 4.5-inch sides encloses (4.5)² = 20.25 square inches.
-- There is no such thing as the least possible area of a rectangle. The longer
and skinnier you make it, the less area it has, even when you keep the same
perimeter. No matter how small I might make the area with the 18-inch string,
you can always come along after me and enclose less area with the same
perimeter, just by making the rectangle longer and skinnier. You can make
the area as small as you want to. You just can never make it zero.
Here are a few examples. ALL of these rectangles have perimeter = 18.
4.5 x 4.5 ...... Area = 20.25
4 x 5 ........... Area = 20
3 x 6 ........... Area = 18
2 x 7 ........... Area = 14
1 x 8 ........... Area = 8
0.5 x 8.5 ..... Area = 4.25
0.3 x 8.7 ..... Area = 2.61
0.1 x 8.9 ..... Area = 0.89
0.01 x 8.99 .. Area = 0.0899
0.001 x 8.999 ... Area = 0.00899
1 micro-inch by 8.999999 inches ... Area = 0.000009 square-inch (rounded)
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So, the difference between the greatest and least possible areas of
the rectangle is:
(20.25) - (the smallest positive number you can imagine) square inches.
