A square give you the most area for the same perimeter.
If the rectangle is composed of 4 squares than it should provide the maximum area because the overall area is equal to the sum of the individual areas.
consider:
x * x = x^2
x + x = 2x
Subtract 1 from the left x and add 1 to the right x and you get:
(x-1) * (x+1) = x^2 - 1
(x-1) + (x+1) = 2x
perimeter remains the same but the area is reduced by 1.
Subtract 2 from the left x and add 2 to the right x and you get:
(x-2) * (x+2) = x^2 - 4
(x-2) + (x+2) = 2x
Perimeter remains the same but the area is reduced by 4.
Square is the maximum area for the same perimeter.
She has 100 meters of fence.
She will need 4 squares.
That's a total of 16 sides.
4 of those sides are on the river bank so she can subtract 4 sides from the requirement to get 12 sides.
6 of those sides are inside the perimeter of the large rectangle so that 2 sides can be collapsed into 1. This reduces the number of sides required by another 3 making the total number of sides required equal to 9
100 meters of fence divided by 9 equals 11.111111111 meters per side.
The exact measurement for each side is (100/9).
With each side measuring (100/9), then the area of each of the squares will be (100/9)^2 = (10000/81)
4 * (10000/81) = (40000/81) = 493.8271605 square meters.
her fence will look like this:
---- ---- ---- ----
| | | | |
| | | | |
That's 9 sides of (100/9) meters each consuming all of the 100 meters of fence.