what should the dimensions of the rectangle be?

Question

marcus has 88 feet of fencing. He wants to build a rectangular pen with the largest possible area. 

Answer 1

Let x = length and y = width

 

Then 2x + 2y = 88.  So, y = 44 - x

 

Area = xy = x(44-x)

 

               = -x² + 44x

 

The graph of the area function is a parabola opening downward.  Since    -x² + 44x =       -x(x-44), the x intercepts are 0 and 44.  By the symmetry of the parabola, the maximum occurs halfway between the x-intercepts.

 

So, the area is maximized when x = 22 ( y = 44 - x = 22)

 

Maximum area when x = 22 ft and y = 22 ft.  

Answer 2

For a given perimeter the rectangle with the greatest area is the square.  Here the square would be 22 x 22 with an area of 484 feet.