Help finding area of a rectangle, excluding one side.

Question

The problem goes as follows: "John has 2,000 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If John does not fence the side along the river, what is the largest area that can be enclosed?" The length is labeled as 2,000-2x and the width is labeled as x on both sides. 

Answer 1

Width = x

 

perimeter = Length + 2*width <--- normally it is twice the length + 2 x width; but not in this problem,

                                                     which states that the length along the river's side is not fenced in.

                                                     So only one length

 

2000 = L + 2x

 

2000 - 2x = L  <---- as given in the problem

 

 

The area is length times width

 

A(x) = (2000 - 2x)x

 

This is a quadratic function and/or a parabola.

 The extreme max/min points occur at the AVERAGE of the zeros.

 The zeros are 2000 - 2x = 0 and x = 0

 

          2000 - 2x = 0 --->  2000 = 2x ----> x = 1000

 

  The zeros are x=1000 and x = 0.

  The average of the zeros are (1000+0)/2 = 1000/2 = 500

 

 The max area occurs at x=500.

 

A quick table verfies this solution.

 

Width x          Length = 2000 - 2x        Area

-------------------------------------------------------

    2                        1996                      3992

  250                    1500                        375000

  500                    1000                        500000

  750                     500                        375000

  999                     2                            1998

 

The max area is in fact 500000 square meters