A box with no top is to be constructed from a piece of...

A box with no top is to be constructed from a piece of cardboard whose length measures 6 in. more than its width. The box is to be formed by cutting squares that measure 2 in. on each side from the four corners and then folding up the sides. If the volume of the box will be $110 \mathrm{in} .^{3}$, what are the dimensions of the piece of cardboard?

Key Concepts

Box Construction from a Flat Sheet

This concept involves designing three?dimensional objects, such as an open-top box, by cutting shapes (usually squares or rectangles) from a flat piece of material and then folding or bending along the edges. It requires understanding how the initial dimensions are modified by removing pieces from the corners and by the subsequent folds, which is crucial for determining the final dimensions and properties of the constructed object.

Volume Calculation of Rectangular Prisms

Understanding how to compute the volume of a box (or rectangular prism) is fundamental when designing containers. The volume is calculated as the product of the length, width, and height. When a box is formed by folding a flat piece, the effective dimensions are reduced by the size and number of cut-outs, and this calculation becomes key to relating the dimensions of the original material to the volume of the finished product.

Algebraic Modeling of Geometric Problems

This concept involves assigning variables to unknown dimensions, setting up equations based on the geometric transformations (such as cuts and folds), and connecting these equations to known formulas (like the volume formula). This approach is essential for translating real-world construction constraints into mathematical models that can be solved.

Solving Quadratic Equations in Applied Contexts

Often, the process of constructing and modeling a box from a flat piece of material leads to quadratic equations. Solving these equations is necessary to determine the unknown dimensions. This concept emphasizes techniques such as factoring, using the quadratic formula, or numerical methods to find valid solutions that make sense in the real-world context of the problem.

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