Arancher wants to fence in an area of 25 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

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