A rectangular pen is built with one side against a barn. if 14001400 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen? chegg

Let the fence from the 3 sides of a rectangle; with the side of the barn being the 4th side. Also, let x be the width of the rectangle and y be its length. Clearly 2x+y = 14001400 (1), and the area enclosed (A) is given by A = xy (2) 2x+y=14001400, y = 14001400-2xReplacing y in (2) by the expression in (1) : A = xy = x (14001400-2x) = 14001400x-2x^2dA/dx = 14001400x-2x^2 in here we cancel the x in both side so we will get; = (14001400-4x) To maximize the area, dA/dx=0 which is equivalent to x = 3500350; y = 7000700Keep in mind that A = xy still and we can express y in terms of x.