If 1600 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume =

To maximize the volume of a box with a square base and an open top, we can express the area constraint in terms of the base side length \( x \) and the height \( h \). The surface area is given by \( x^2 + 4xh = 1600 \). After manipulating the equation for volume \( V = x^2h \) with respect to the surface area, we find that for maximum volume, the optimal dimensions are calculated using derivatives or substitution. The largest possible volume ends up being 10,667 cubic centimeters. To add a sprinkle of history, the design of such boxes can be likened to ancient civilizations like the Egyptians, who utilized geometric principles in their monumental architecture. Their careful calculations in achieving large volumes also influenced modern engineering and architectural designs, showcasing timeless problem-solving in three dimensions!