Shortest length: A rancher wants to fence in an area of 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?

Question

Shortest length: A rancher wants to fence in an area of  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?

anonymous · Answer

How many square feet? It didn't show.

anonymous · Answer

woops! 500,000!

anonymous · Answer

Your primary equation will be to describe the perimeter of the fenced in area, which for a rectangle would be P=2x+2y. However, because of the divide down the middle, it is instead P=3x+2y (it will be equal in length to one of the sides). You need to find an equation to relate x and y so that you can substitute one for the other, and this can be done with your area. The area equals side x times side y, so xy=500,000.

anonymous · Answer

Substituting for one variable or the other gives you a "good" primary equation - one that is written in one variable. Derive with respect to that variable, and you will be able to find your critical numbers.

anonymous · Answer

i thought it would be P=x+3y, since it's a wall, and the rancher can only fence the top, and the sideways, and the middle of the wall?

anonymous · Answer

I had assumed that there were no barriers on a side. One could reduce the number of sides of the perimeter if, say, there were a river on one side of the field. However, barring anything that will serve as a wall already, my equation stands.

anonymous · Answer

Another possible complication of a pre existing wall would be whether the equation would be P=2x+2y or P=x+3y, because the question did not state which side the divide in the middle was to be parallel to. Therefore, I assumed that there were no pre existing walls.

anonymous · Answer

i see! that made sense to me thanks!