Question 1074737
.
The solution by "josgarithmetic" is   {{{highlight(WRONG)}}}.


Below find the correct solution.


<pre>
The area of the garden, under the given condition, is

A = y*(80-2y) = -2y^2 + 80y.


Referring to the general form of a quadratic function

A = ay^2 + by +c,

the function have a maximum at y = {{{-b/(2a)}}}, which is y = {{{-80/(2*(-2))}}} = {{{90/4}}} = 20.


Thus the maximum is achieved at y = 20 m, and the maximal value of the quadratic function (of the area) is 

A = -2*20^2 + 80*20 = -2*400 + 1600 = 800 square meters.


<U>Answer</U>.  The dimensions of the garden are 20 m x 40 m. Its area is 800 square meters.
</pre>


------------------------
The statement 

<pre>
     At given perimeter, the area of a rectangle is maximal if and only if the rectangle is a square
</pre>

is true if the PERIMETER (the entire perimeter consisting of four sides) is constrained.


In the given problem we have ANOTHER/DIFFERENT situation.  (with which "josgarithmetic" is unfamiliar, due to his mathematical illiteracy).
------------------------


{{{graph( 330, 330, -5.5, 50.5, -100.5, 1000.5,
          -2x^2 + 80x
)}}}


Plot A = {{{-2y^2 + 80y}}}



On finding the maximum/minimum of a quadratic function see the lessons

&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-farmer-planning-to-fence-a-rectangular-area-along-the-river--to-enclose-the-maximal-area.lesson>A farmer planning to fence a rectangular area along the river 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/Using-quadratic-functions-to-solve-problems-on-maximizing-profit.lesson>Using quadratic functions to solve problems on maximizing revenue/profit</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic "<U>Finding minimum/maximum of quadratic functions</U>". 



In the list of lessons, one is marked by the (*) sign.
It is your prototype/sample.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;H&nbsp;a&nbsp;p&nbsp;p&nbsp;y &nbsp;&nbsp;l&nbsp;e&nbsp;a&nbsp;r&nbsp;n&nbsp;i&nbsp;n&nbsp;g &nbsp;!&nbsp;!