A rancher wants to fence in an area of 500000 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?

let the sides by x and y. then xy = 500000, so y = 500000/x

the length you want to minimize is f(x) = 3x + 2y (because of the extra bit down the middle)

f(x) = 3x + 1000000/x

to find the optimal length, solve df/dx = 0

df/dx = 3 - 1000000/x^2 = 0

i.e., solve 3 = 1000000/x^2

x^2 = 1000000 / 3

x = 1000 / sqrt(3)

shortest total length is f(1000 / sqrt(3))

= 3000 / sqrt(3) + 1000 sqrt(3)