Question 647779
Material available = 100ft (the constraint)
Need a rectangle of only three sides - either 2 length with 1 breadth/width or 2 breadth/width with 1 length.

Let x = length and y = width denote the sides of the rectangle (assuming 2 length and 1 width)


=> The perimeter is {{{(x + x) + y = 100}}} .... eqn 1 
and the Area is {{{x * y}}} .................... eqn 2


Then, we make y the subject in eqn1 to substitute into eqn2
eqn1 => {{{(x+x)+y=100}}}
        {{{2x + y = 100}}}
         {{{y = 100 - 2x}}} .....eqn3
therefore, eqn3: {{{y = 100 - 2x}}}  into eqn2:
=>  {{{x * y}}}
  = {{{x * (100 - 2x)}}
   {{{100x - 2x^2}}} ....eqn4
However, since eqn4: {{{100x - 2x^2}}} or {{{-2x^2 + 100x}}} is an equation of a parabola curve, thus, we calculate the maximum point for x using the formula {{{(-b)/2a}}}


From the eqn4, a = -2 and b = 100
so, {{{(-100)/(2*-2)}}}
  = {{{(-100)/(-4)}}}
  = 25 (hence, x, one side of rectangle (the length) is 25ft)


Given x = 25ft, we find the value for other side(y)
From eqn3: {{{y = 100 - 2x}}}
           {{{y = 100 - 2*25}}}
           {{{y = 100 - 50}}}
           y = 50 (hence, y, the other side (the width) is 50ft)


The maximum area can be achieved with 2 sides = 25ft and 1 side = 50ft
Area = {{{25 * 50}}}
Area = {{{1250ft^2}}}


The graph of  {{{-2x^2 + 100x}}}
{{{graph(300,200,-30,70,-2000,2000,-2x^2+100x)}}}


Hence, from the graph above, it can be clearly seen that the maximum point of the graph lies between 20 and 30 on the x-axis which is 25ft.