a farmer can afford 8000 square meters of fence and would like to come up with the best way to use this fence, such that:

Question

- he wants all the pens to be of the same rectangular shape.
- he wants to give each calce of his farm as much space as possible.
- he needs an easy access to each calf
he wants 16 eparate spaces.

Answer 1

This depends on how the pens are arranged. If the pens are all separate, then more fence is used than if they share sides.

Also, fence is not measured in square meters, but just meters.

16 pens in a line of width x and length y, sharing the walls of width x, use up

17x+32y meters of fence.

Make a double row sharing long sides y as well, and you use

18x+24y meters.

Have 16 separate pens, and you use

16(2x+2y) meters

Pin down the problem some more, and maybe we can get somewhere.

Answer 2

For a given perimeter, the rectangle that encloses the maximum area using one side as a building, river, fence, etc. or an adjacent section of fence, is in the ratio of 2/1. Assuming each of the 16 grazing areas is in the length to width ratio of 2/1:

Let the length of the 16 enclosed areas be L, the width of the 16 enclosed areas be W, the length of a single area is l and the length of a single area is W.

L = 16w
W/w = 2
L = 2W

17W + 2L = 8000
Using W/w = 2 and L = 16w
34w + 32w = 8000
66w = 8000
w = 121.212
W = 242.424
L = 16(121.212) = 1939.36

17(242.424) + 2(1939.36)
4121.208 + 3878.72 = 8000 meters.