Among all rectangles that have a perimeter of 144. find the demensions of the one whose area is largest

Question

I don't know how to set this up and I am needing help. My professor did not tell me how to do this nor does my book

Answer 1

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.

 

 

 

                  

Answer 2

The answer is a square. I don't remember how to prove it mathematically.

 

So the dimensions are perimeter/4 = 144/4 = 36 for both length and width.