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 here?

To minimize the length of the fence, we can denote the length of the rectangle as ( l ) and the width as ( w ). The area constraint gives us ( l \times w = 500,000 ). The total length of the fence, including the dividing fence, is ( 2l + 3w ). To minimize this, we can express ( w ) in terms of ( l ) and use calculus or substitution, ultimately finding that the optimal dimensions are ( l = 1000 ) feet and ( w = 500 ) feet, resulting in a minimum fence length of 3,000 feet.