Applied optimization

Question

A rancher wants to fence in an area of 2,000,000 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?

  

Answer 1

Let x = length of dividing fence and y = length of one side of fence perpendicular to the divider

Total length of fence = L = 3x + 2y

Since xy = 2000000, y = 2000000/x.

So, L = 3x + 4000000/x

L' = 3 - 4000000/x² = (3x² - 4000000) / x²

L' = 0 when 3x² = 4000000. So, x = 1154.7 ft and y = 2000000/x = 1732.1 ft

L = 3x + 2y = 6928.3ft