Maximizing area: rectangular fence

Question

A farmer wants to fence a rectangular area by using the wall of a barn as one side of the rectangle and then enclosing the other three sides with 160 feet of fence. Find the dimensions of the rectangle that give the maximum area in side. 

Answer 1

Let y = length of side parallel to the wall

     x = length of each side perpendicular to the wall

 

Then, y+2x = 160.  So, y = 160-2x

 

A = area = xy = x(160-2x) = -2x²+160x, 0 < x < 80

 

The graph of the area function is a parabola opening downward.  The maximum occurs at the x-coordinate of the vertex                                       (x = -160/(2(-2)) = 40)

 

To maximize the area, each side perpendicular to the wall should have length x = 40 ft and the side parallel to the wall should have length     y = 80 ft.