A woodcraft artisan’s most popular items are her pairs of cedar nesting boxes. Her craftsmanship is so precise that one open-topped box fits perfectly inside another. The outer box is a cube with side length [latex]x[/latex] cm. The box that nests inside of it is a rectangular prism with its open face lying flush with the open face of the outer box. Both boxes, including their bottoms, are made with cedar that is [latex]1[/latex] cm thick. The outer box (left) and the inner box (right) are depicted below.

To help her customize her nesting boxes to her patrons’ volume needs, the artisan uses polynomial expressions to represent the various volumes involved.