A rancher wants to fence in an area of 2500000 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?

Lets say that,

x = width of rectangle

y = length of rectangle

The area of the rectangle is given as:

A = x * y = 2500000 ft^2

Rewriting in terms of y:

y = 2500000 / x

The length of fencing needed is equal to the perimeter of rectangle plus the middle fence:

L = 2 (x + y) + x = 3x + 2y

Substituting the value of y:

L = 3x + 2 (2500000/x) = 3x + 5*10^6*x^ (-1)

The minima can be obtained by taking the 1st derivative of the equation then equating dL/dx = 0:

dL/dx = 3 - 5*10^6*x^ (-2)

3 - 5*10^6*x^ (-2) = 0

x^-2 = 6*10^-7

x = 1291

Calculating for y:

y = 2500000 / x = 2500000 / 1291

y = 1936.48

Therefore the shortest length of fence needed is:

L = 3x + 2y = 3 (1291) + 2 (1936.48)

L = 7746 ft