Question 1189543
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            Instead of solving your problem, I'll give you the solution to a TWIN problem,

            to give you an opportunity to learn the method and the subject.



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A farmer plans to fence a rectangular grazing area along a river with  300 yards of fence. 
What is the largest area he can enclose?


<B>Solution</B>


<pre>
Since one side is the river, the rectangle's fence perimeter will be

    L + 2W = 300.

Hence, L = 300 - 2W.


    Area = Length * Width.


Substitute (300-2W) for L:

    A = W(300 - 2W)

    A = -2W^2 + 300W.


It 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 = {{{-300/(2*(-2))}}} = {{{(-300)/(-4)}}} = 75.


So, W = 75 yards is the width for max area.


Then the length is  L = 300 - 2W = 300 - 2*75 = 150 yards.


Find the maximum area. It is

    A = L*W = 150*75 = 11250 square yards.


The plot of the quadratic function for the area is shown below:  y = area and x = width.


{{{ graph( 300, 200, -50, 200, -1000, 12000, -2x^2 + 300x) }}} 
</pre>


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/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>