Question 1139604
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Since one side is the river, the rectangle's fence perimeter will be

L + 2W = 160.

Hence, L = 160 - 2W.

Area = Length * Width.

Substitute (160-2W) for L:

A = W(160 - 2W)

A = -2W^2 + 160W.

This is a quadratic function. It has the maximum at x = -b/(2a), 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).


For our quadratic function the maximum is at

W = {{{-160/(2*(-2))}}} = {{{(-160)/(-4)}}} = 40.

So, W = 40 meters is the width for max area.


Then the length is  L = 160 - 2W = 160 - 2*40 = 80 meters.


Then the maximal area is L*W = 80*40 = 3200 square meters.


The plot of the quadratic function  y = - 2x^2 + 160x  for the area is shown below:  y = area and x = width.



    {{{ graph( 300, 200, -20, 100, -1000, 4000, -2x^2 + 160x) }}} 


    Plot y = -2x^2 + 160x.
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