Question

A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
smaller value? and larger value?

Answer 1

Use half the fencing in each direction: 3 segments in one direction, 2 in the other. 

W * L = 24 million 
3 W = 2 L 

W * 3 W / 2 = 24 million 
W^2 = 16 million 
W = 4000 
L = 6000 

This shortcut (3 W = 2 L) 
is derived from total fencing = 3W + 2L 
and W*L = 24 million 
total fencing = 3W + 2 * 24 million / W 
and taking the derivative, setting to 0, and solving for W