Question 1163079
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<pre>

Since one side is just fenced, the total length of the three other sides of the rectangle, W, L an W is

L + 2W = 180 meters.

Hence, L = 180 - 2W meters.


    Area = Length * Width.


Substitute (180-2W) for L:

    A = W(180 - 2W)       (1)

    A = -2W^2 + 180W.


It is a quadratic function. It has the maximum at x = -b/(2a),  where "a"  is the coefficient at the quadratic term 
and  "b"  is the coefficient at the linear term, according to the general theory.

    (See the lessons
     
         - <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

         - <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

     in this site).


In your case, the maximum is at


    W = {{{-180/(2*(-2))}}} = {{{(-180)/(-4)}}} = 45.


So,  W = 45 meters is the width of the rectangle for the max area.


Then the length is  L = 180 - 2W = 180 - 2*45 = 90 meters.


So, the dimensions of the rectangle are 90 m and 45 m.


The max area is 


    A = L*W = 90*45 = 4050 square meters.    <U>ANSWER</U>.
</pre>

Solved.


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My other lessons in this site on finding the maximum/minimum of a quadratic function are 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/A-rectangle-with-the-given-perimeter-which-has-the-maximal-area-is-a-square.lesson>A rectangle with a given perimeter which has the maximal area is a square</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/A-farmer-planning-to-fence-a-rectangular-garden-to-enclose-the-maximal-area.lesson>A farmer planning to fence a rectangular garden to enclose the maximal area</A> (*)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/A-rancher-planning-to-fence-two-adjacent-rectangular-corrals-to-enclose-the-maximal-area-.lesson>A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Finding-the-maximum-area-of-the-window-of-a-special-form.lesson>Finding the maximum area of the window of a special form</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/Using-quadratic-functions-to-solve-problems-on-maximizing-profit.lesson>Using quadratic functions to solve problems on maximizing revenue/profit</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/OVERVIEW-of-lessons-on-finding-the-maximum-minimum-of-a-quadratic-function.lesson>OVERVIEW of lessons on finding the maximum/minimum of a quadratic function</A>



The most relevant to your problem is the lesson marked (*) in the list.



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My notice after reading the post by @Theo.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The original formulation in the post is absolutely and crystally clear.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The interpretations that @Theo tries to make, &nbsp;to discuss and to use, &nbsp;all are irrelevant and do not fit to the problem.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Simply ignore his post, &nbsp;for your safety.