SOLUTION: a farmer has 100 meters of fence to enclose the maximum area possible. she wishes to use a river bank as a natural boundary for one side of the rectangular garden that she desires

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Question 239763: a farmer has 100 meters of fence to enclose the maximum area possible. she wishes to use a river bank as a natural boundary for one side of the rectangular garden that she desires to construct. some of the fence will also be used to divide the rectangular into 4 equal rectangular sections
a)what is the maximum area that cna be enclosed
b) what dimensions, lenght and widht , produce the maximum area?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
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.