A farmer wants to make a rectangular field with a total area of 3200^2. It is surrounded by a fence. It is divided into 3 equal areas by fences. What is the shortest total length of fence with which this can be done?

18133.333... units

Step-by-step explanation:

the shortest circumference for a given area based on rectangles is a square (all sides of equal length).

we cannot split a square completely into 3 equal area squares (only 4).

the best approach is then to split this main square into 3 equal area rectangles (not equal rectangles).

area of square 3200² = 10240000

a third of that area is 3413333.333...

now, splitting the main square into 3 equal "strips" of land creates 3 narrow and long rectangles, which is bad for minimizing the circumference. the best solution is to get as closely as possible to a square shape.

that we get by splitting one side in half and go in as far as the land area allows, which would be

half a side = 1600

3413333.33.../1600 = 2133.33...

so, we get 2 areas of 1600 × 2133.33...

and one area of (3200-2133.33...)×3200 = 1066.66...×3200

the length of fence needed is then

around the main square = 4×3200 = 12800

plus one length of fence between the first 2 parts : 2133.33...

plus one intensive long side between the first 2 parts and the remaining third part : 3200

in total : 18133.33...

to check : the three slices would require the 12800 plus 2 long lines inside between the 3 slices : 2×3200

in total : 19200

and an attempt to simply build one long strip of 3 equal area squares next to each other :

the side length of one such square is

sqrt(341333.33...) = 1847.520861...

as fence we would need 6 times that side length for the top and the bottom of the long strip, 2 times that side length for the sides left and right if the strip, and 2 times inside as additional delimiters between the 3 squares.

in total : (6+2+2) × 1847.520861... = 10×1847.520861...

= 18475.20861...

this is all larger than the original solution.