Need helping setting up equation to this problem

Dec 2009
A farmer wants to enclose a rectangular field by a fence and divide it into two smaller rectangular fields by constructing another fence parallel to one side of the field.
The famer has 3000 yards of fencing. Find the dimensions of the fields so that total enclosed area is a maximum.

What equation would I use to find the length and how do I know that it is the maximum area?



MHF Helper
Apr 2005
Let "w" be the width of the field and "b" the breadth. Then the area enclosed is bw. The total fencing used, so far, is b+ w+ b+ w= 2b+ 2w. If the third fence is parallel to the width, and has length w ("w" and "b" are interchangeable so this is no restriction), then the area is still bw but the total fencing used is 2b+ 2w+ w= 2b+ 3w. Since you have 3000 yards of fencing, you can set 2b+ 3w= 30000 and can the solve for b= (3000- 3w)/2= 1500- (3/2)w. Then the total area is still (1500- (3/2)w)(w)= 1500w- (3/2)w^2.

I don't know what methods you have learned to fimd minimum values for a given function, but because this is posted in "algebra", I would recommend "completing the square".