Calc word problem

Question

A rancher wants to fence in an area of 1,000,000 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 and y be the dimensions of the rectangle

 

If x is the length of the fence down the middle, then

 

L = total length = 3x + 2y

 

Since xy = 1,000,000, y = 1,000,000/x

 

So, L = 3x + 2,000,000/x

 

L' = 3 - 2,000,000/x²

 

L' = 0 if (3x² - 2,000,000) / x²  = 0

 

3x² = 2,000,000

 

  x² = 666,666.666666

 

  x = 816.5 ft     y = 1,000,000/x = 1224.74 ft

 

L = 3x + 2y ≈ 4899 ft