The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal-size squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches. Use the table feature of a graphing calculator to do the following. Round to the nearest hundredth.
b. Determine when the volume of the box will be greater than 40 in.³.

Verified step by step guidance

1. Identify the dimensions of the original piece of paper: length = 11 inches and width = 8.5 inches.
2. Express the dimensions of the box after cutting out squares of side length \(x\) from each corner. The new length will be \$11 - 2x\(, the new width will be \)8.5 - 2x\(, and the height will be \)x$.
3. Write the volume \(V\) of the box as a function of \(x\): \(V(x) = x \times (11 - 2x) \times (8.5 - 2x)\).
4. Set up the inequality to find when the volume is greater than 40 cubic inches: \(x \times (11 - 2x) \times (8.5 - 2x) > 40\).
5. Use the table feature of a graphing calculator to evaluate \(V(x)\) for various values of \(x\) and determine the range of \(x\) values for which the volume exceeds 40, rounding your answers to the nearest hundredth.

Key Concepts

Polynomial Modeling of Volume

This concept involves expressing the volume of the box as a polynomial function of the variable x, the side length of the cut squares. The volume is calculated by multiplying the length, width, and height of the box, which depend on x, resulting in a cubic polynomial that models how volume changes with x.

Domain Restrictions and Feasibility

Since x represents the side length of the squares cut from the corners, it must be positive and less than half the smaller side of the paper (less than 4.25 inches). Understanding these domain restrictions ensures the model reflects a physically possible box and avoids invalid or negative dimensions.

Using a Graphing Calculator Table to Solve Inequalities

The table feature on a graphing calculator helps evaluate the polynomial volume function at various x-values. By examining these values, students can identify when the volume exceeds 40 cubic inches and approximate the solution to the nearest hundredth, facilitating solving inequalities involving polynomial functions.