# A long math problem

A farmer has 76 feet of fencing and wants to build a rectangular pen for his chickens. What should the dimensions of the pen be if h wants it to have the greatest area possible?

Hi Whisper, This problem isn't as long as you might think. You should try out some different dimensions (which add up to 76 of course) and work out the areas. It's trial and error but you should fairly quickly start to notice a pattern. Let me know if you need any more help.
sinsua
13 March 2012
wouldn't it be 35 x 41
aliyu123
15 March 2012
We know the area for a rectangle is: Where is area, is width, is length.
theo.cushion
16 March 2012
You could solve it iteratively as sinsua suggests. Sometimes this is a good way to check a result as well. However, we can actually solve it using some simple techniques.
theo.cushion
16 March 2012
We also know that the perimeter of a rectangle is: Where is perimeter
theo.cushion
16 March 2012
Now we are interested in maximising , while keeping .
theo.cushion
16 March 2012
We cannot solve yet as there are too many unknowns, but maybe we can get further with the other equation: Now we could rearrange this to make either or the subject of the forumla. Let's go for .
theo.cushion
16 March 2012
A quick recap, we now have 2 equations:
theo.cushion
16 March 2012
We can use substitution now to combine our top equation into the bottom one: Whisper: Do you recognise this equation?
theo.cushion
16 March 2012
Is the problem really that the questioner has misled you by the use of the term "rectangular"? Is there another shape that you may know under another name but does in fact satisfy the requirements for "rectangle"?
ianmoth
16 March 2012
ianmoth: I suspect, that this is an exam question and part of a standard syllabus where a student needs to demonstrate they are comfortable with quadratics and completing the square in an applied maths problem. Although you are right, there is certainly an easier way if you don't have to go through first principals!
theo.cushion
16 March 2012