Arancher wants to fence in a rectangular area of 14900 square feet in a field and then divide the region in half with a fence down the middle parallel to one side. what is the smallest length of fencing that will be required to do this?

Suppose the length of the fence perpendicular to the central partition is x. Then the total length of the fence (L) for enclosed area A is
.. L = 2x +3*A/x
The derivative of this with respect to x is
.. L' = 2 -3A/x²
L will be minimized when this is zero.
.. 0 = 2 -3A/x²
.. x² = 3A/2
.. x = √(3A/2)

For this value of x, the total length of fence is
.. L = 2√(3A/2) -3A/√(3A/2)
.. L = 4√(3A/2) = 2√(6A)

For this problem, A = 14,900 ft², so
.. L = 2√(6*14,900 ft²) ≈ 597.997 ft

The smallest length of fence required to fence the rectangular area is 598 ft.