Let x = length and y = width
Then, 2x+2y = 144. So, x+y = 72
y = 72 - x
Area = xy = x(72-x)
= -x2 + 72x
The graph of the area function is a parabola opening downward, with maximum occurring at the x-coordinate of the vertex. The x-coordinate of the vertex lies halfway between the x-intercepts.
To find the x-intercepts, solve the equation x(72-x) = 0. We see that the x-intercepts occur when x = 0 and x = 72.
So, the area is maximized when x = 36 and y = 72 - x = 36.
The rectangle whose perimeter is 144 and which has largest possible area is a square with side length of 36.