An open box is made from an \(s\text{-}by\text{-}s\) square sheet by cutting \(x\text{-}by\text{-}x\) squares from the corners. Use the diagram to build a \(\text{one}\text{-}\text{variable}\) volume model with its positive feasible interval, then find the optimizer, box dimensions, and maximum volume.

Translate the cut-and-fold geometry before optimizing: the cut size becomes the height, while each base dimension loses that amount at both ends. Multiply height by the resulting base area to get one variable, and restrict it so every physical dimension remains positive. Compare interior critical points with the zero-volume boundaries, then substitute the optimizer back to recover all dimensions.