Question 1146865
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            Regarding this post,  I have two notices.



1.    Your formulation is  INCORRECT.


          The question should ask about the  MINIMUM  length of the fencing ---- NOT about the maximum length. 

          The maximum length  DOES  NOT  EXIST.    You can make your enclosure longer and narrower,  by keeping the same area.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The correct formulation is &nbsp;<U>THIS</U> :


<pre>  
           What is the {{{highlight(minimum)}}} fencing length needed to construct a rectangle enclosure 
           containing 1800 ft^2 using a river as a natural boundary on one side? 
</pre>


2.  &nbsp;&nbsp;The "solution" by @josgarithmetic is &nbsp;&nbsp;TOTALLY &nbsp;&nbsp;WRONG, &nbsp;starting from its third line to the end.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So you better simply &nbsp;IGNORE &nbsp;it.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below find my <U>correct</U> solution.



<pre>
xy = 1800              (1)

x + 2y -----> minimize        (x is the length along the river)



So your task is to minimize (x+2y) under the given condition/restriction  (1).



From (1),  x = {{{1800/y}}},  so we need to minimize the function  f(y) = {{{1800/y + 2y}}}.


The derivative  f'(y) = -{{{1800/y^2}}} + 2.


To find the minimum of f(y), equate its derivative to zero


    -{{{1800/y^2}}} + 2 = 0


    {{{1800/y^2}}} = 2

    {{{y^2}}} = {{{1800/2}}} = 900

    y = {{{sqrt(900)}}} = 30.


<U>ANSWER</U>.  The minimum fencing is at y = 30 ft perpendicular to the river and x = {{{1800/y}}} = {{{1800/30}}} = 60 ft along the river.
</pre>

Solved.