Optimizing a Rectangular Pen: Maximizing Area with a Fixed Fence and a River

① This GeoGebra example demonstrates a classic optimization problem: a farmer uses 200 meters of fencing to build a rectangular pen along a straight river, where the river forms one side, and seeks the dimensions that maximize the enclosed area. ② The core mathematical concepts include quadratic functions, parabola vertex, derivative-based extremum finding, and constrained optimization, along with modeling area as a function of one variable using the perimeter constraint. ③ Construction highlights: commands such as Line, Polygon, Circle, Intersect, OrthogonalLine, Segment, Function, and Derivative are used to build the rectangle and auxiliary lines; the area function is defined via Function, its derivative is computed, and the extremum point is marked; dynamic sliders or curves allow exploring how area changes with dimensions, involving objects like points A–H, segments a, b, c, polygons poly1 and poly2, and the Area measurement. ④ This is suitable for high school or introductory college mathematics, serving as an interactive tool for teaching derivative applications in optimization, or as a visual aid in mathematical modeling and pre-calculus courses.