Question

A rancher wants to fence in a rectangular area of 23000 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?

Answer 1

If the length and width are x,y, then we want to minimize

f = 3x+2y subject to
xy = 23000

so, y=23000/x, and we want the minimum of

f = 3x+2(23000/x)
df/dx = 3 - 46000/x^2
df/dx =0 at x = 20/3 √345)
f = 40√345