Among all rectangles that have a perimeter of 130, find the dimensions of the one whose area is largest.

Question

Among all rectangles that have the perimeter of 130, find the dimensions of the one whose area is largest. Write answer in reduced terms 

Answer 1

Ralph's answer is correct, but let's see if we can show that his rule gives the correct answer.

 

The perimeter of a rectangle is 2(L + W).   In this case we know that the perimeter is 130.

 

So 2(L+W) = 130.   Solving for L  gives  

 

                         L = 65 - W.

 

The area A of a rectangle is given by A = LW.   We can get an expression for the area in terms of W alone by replacing L with our expression for L obtained above.

 

                       A = (65-W)(W) =  65W - W²

 

The equation above can be recognized as a parabola.   We can locate the coordinates of the vertex by completing the square as follows:

 

                      A -  (65/2)² = -(65/2)² + 65W - W²

 

                      A - 32.5² =  -1(W - 32.5)²

 

Once the equation is in the form  y-k = a(x-h)², we can recognize that the graph is an inverted parabola (from the -1) and that the vertex is the location of the maximum area.   The coordinates of the vertex are (h, k) or (32.5, 32.5²) 

 

So the maximum area occurs when W = 32.5.   Since L = 65 - W,  L = 65 - 32.5 = 32.5.   

Answer 2

A rectangle with all sides equal (square) with a fixed perimeter will produce maximum area.

 

so,

 

130/4= 32.5