Arancher wants to fence in an area of 3 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. what is the shortest length of fence that the rancher can use?

Let X and Y be the two dimensions.
Area will be X times Y.
XY = 3000000
Y = 3000000 / X
Perimeter  = 2x + 3y
substituting the values:
Perimeter = 2x + 3*3000000 / x
                = 2x + 9000000 / x
 
2 = 9000000 / x^2 
x^2 = 9000000 / 2 = 4500000 

x = sqrt(4,500,000)
   = 2121.32
   = 1000 sqrt (4.5)

y = 3000000 / 2121.32
   = 1414.2
   = 1000 sqrt(2) 

so the dimensions will be 2121.32 by 1414.2 .