a$.$ A rectangular pen is built with one side against a barn. If $100$ m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen? b$.$ A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of $25$ m$^2.$ What are the dimensions of each pen that minimize the amount of fence that must be used? a$.$ To maximize the area of the pen, the sides perpendicular to the barn should be m long and the side parallel to the barn should be m long. $($Type exact answers, using radicals as needed.$)$