An open box is to be made from a rectangular piece of...

An open box is to be made from a rectangular piece of cardboard 20 inches $\times 30$ inches by cutting out identical squares of area $x^{2}$ from each corner and turning up the sides (see figure). Express the volume $V$ of the box as a function of $x$.

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Key Concepts

Transformation of Dimensions

When creating a box by cutting out squares from each corner of a rectangular piece, the effective length and width of the base are reduced by twice the side length of the square cutouts. This concept is fundamental in establishing the new dimensions after the cuts, as each dimension loses material from both ends.

Volume of a Box

This concept refers to calculating the volume of a three-dimensional box by multiplying its length, width, and height. In problems involving an open-top box formed by cutting and folding a flat piece of material, the volume is determined by the product of the modified base dimensions and the height, which is created by the size of the cutout.

Algebraic Modeling

Algebraic modeling involves expressing real-world scenarios using mathematical expressions and functions. In this context, the volume of the box is represented as a function of the variable x, which is the side length of the cutout squares, allowing for analysis and optimization by composing the resulting algebraic expression.

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