A rectangular open-top box is to be constructed out of an...

A rectangular open-top box is to be constructed out of an 8.5 -inch by 11 -inch sheet of thin cardboard by cutting $x$ -inch squares out of each corner and bending the sides up, as in Figures 39 and 40 in Example $6 .$ What size squares to two decimal places should be cut out to produce a box with a volume of 55 cubic inches? Give the dimensions to two decimal places of all possible boxes with the given volume.

Key Concepts

Polynomial Equation Formation and Solving

Once the volume function has been established by substituting the appropriate dimensions, setting it equal to a given value produces a polynomial equation, often of cubic degree in x. Solving this equation requires either analytical or numerical methods to find the possible x values that yield the desired volume. This process involves isolating valid roots and identifying which solutions are practical based on the context of the problem.

Physical Constraints in Geometry Problems

This key concept emphasizes that not all algebraic solutions may be admissible in a real-world context. For the box construction problem, the cut-out size x must be positive and small enough so that when doubled and subtracted from the original sheet dimensions, the remaining length and width are still positive. These physical constraints ensure that the mathematical model stays true to the geometric reality, filtering out extraneous or impossible solutions.

Box Construction from a Sheet

This concept deals with altering a flat sheet by cutting out equal-sized squares from each corner and folding up the resulting flaps to create an open-top box. The process involves understanding how the initial dimensions of the sheet are reduced by the cut-out squares, thereby affecting the lengths, widths, and ultimately the volume of the box. It is a fundamental idea in geometric modeling where a two-dimensional shape is transformed into a three-dimensional object.

Volume Function for Constructed Boxes

The volume function is derived from the dimensions of the box that result from the construction process. With squares of side length x removed from every corner, the new length and width become (original length ? 2x) and (original width ? 2x) respectively, while the height of the box is x. The overall volume is modeled as V = x*(width ? 2x)*(length ? 2x), which is a crucial step in connecting the geometry of the box to an algebraic expression that can be manipulated to meet design criteria (such as a specified volume).

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