



A large
farm field is next to a river.



The
farmer has 900 feet of fence 


and
wants to fence in as much area as possible 


as a rectangular fields
next to the river. 


Because
the river is there, only 3 sides need to be fenced. 








How
big an area can the farmer fence in? 





Here's
how it goes. 


The
sides on the left and right are the same length 


We
don't know what that length is, 


so
we'll call each of them X. 











The
other side is some other length. 


We
don't know how long it is either. 


What
we do know is that the farmer has 900 feet of fence total. 


The
left and right sides use up 2X. 


What's
left for the other side is: 





900  2X






So
now we have the lengths of all three sides, sort of ... 








The
area that is enclosed is equal to length times width. 


We
can call the length X and the width 900  2X 


So: 



Area =
length x width 



Area =
X(900  2X) 



Area =
2X^{2} + 900X 





Hey!
This equation is one of those parabola things! 


Yep!
It sure is ... 


Because
the "A" is 2, we know that this parabola opens down ... 








From
the stuff on the last page we also know something else. 





The
vertex of the graph is at: 








Area
= 2X^{2} + 900X






A
2 B = 900 C = 0



(if
one of the numbers is missing, then it is zero)












Now
we can use this value for X to figure out the maximum area. 






Area =
2(225)^{2} + 900(225) 



Area =
101,250 + 202,500 



Area =
101,250 





We
get X = 225, Y = 101,250 





So
the maximum area is 101,250 square feet (about 2 1/2 acres). 


the
measurements of the fenced field are: 








copyright 2005 Bruce Kirkpatrick 
