A farmer has 200 feet of fencing to enclose a rectangular pen along the side of a barn, so fencing is needed on only three sides. What is…

Let $x$ be the length of each of the two sides perpendicular to the barn. The side parallel to the barn is $200-2x$, so the area is $A=x(200-2x)=200x-2x^2$. This parabola opens downward, and its vertex is at $x=-\frac{200}{2(-2)}=50$. Then $A(50)=50\cdot100=5000$ square feet, choice J. Choice F ($2{,}500$) is $50^2$, which treats the pen as a square. Choice H ($10{,}000$) and choice G ($20{,}000$) are too large; at $x=100$ the area would be zero. The maximum occurs when the parallel side is twice the perpendicular sides.