Expert: Paul Klarreich - 5/29/2007

Question
A rectangular field is surrounded by 240m of fencing on three sides. the other side is a hedge where no fencing is required. find the dimensions x and y that command the maximum area of the field. (y being length, x being width).

Answer
Questioner:   alex
Category:  Calculus
Question:  A rectangular field is surrounded by 240m of fencing on three sides. the other side is a hedge where no fencing is required. find the dimensions x and y that command the maximum area of the field. (y being length, x being width).
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Hi, Alex,

This is one of the classic 'Farmer Brown' problems, which all say:  Farmer Brown has XXX feet/meters of fencing to make a chicken coop under the following conditions, like one side is a wall or hedge, or he needs a fence in the middle, etc.  

In this case, you (or Farmer Brown) need two x's of fencing plus one y, because you can use the wall/hedge as one boundary.

The scheme for these is:

1. Identify the quantity to be max/min-ized.
2. Express it in terms of ONE variable.  
2A. If you have two variables, find some condition that lets you eliminate all but one.
3. Differentiate, set the derivative = 0, find your critical points, which include endpoints of the "interval of meaninfulness".
4. Pick the best one.

Let A = the area of the field.
Then  A = xy.
But  2x + y = 240,  so  y = 240 - 2x

A = x(240 - 2x) = 240x - 2x^2

A' = 240 - 4x

Set  240 - 4x = 0,   x = 60

Interval of meaningfulness:  x >= 0 and  x >= 120.

If x = 60,  y = 240 - 120 = 120, and  A = 7200  BEST
IF x = 0, A = 0   NOT BEST
If x = 120, y = 0, and A = 0.   NOT BEST

Max area is 7200, dimensions are  60 by 120.