Your answer looks correct to me. You can simplify your final answer a little bit by adding the fractions.
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)