Problem 2

 

Suppose a rancher wants to enclose an area along the side an existing fence.  If the enclosed area is to be rectangular and the rancher has a fixed amount of fencing material to make the other three sides, what ratio of length to width will result in the largest possible area? 

[Problem submitted by Vin Lee, LACC Associate Professor of Mathematics.]

 

 

 

 

 

 


Solution:

 

Let p be the total length of the 3 new sides of the rectangle, x the length of the side parallel to the old fence, and y the length of each of the 2 sides perpendicular to the old fence.  Then

                          

                       

                        Area = xy

                               

                               

This is a quadratic equation in standard form.  If it is graphed with Area on the vertical axis and y on the horizontal axis, then the graph is a downward opening parabola whose maximum value of Area occurs at the vertex when .  So, 

                       

                          

                        .

So, the maximum area occurs when the ratio of length to width is  = 2.