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 cor-ner 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. Find the maximum volume of the box.

Verified step by step guidance

1. <Step 1: Define the dimensions of the box.> The original piece of paper is 8.5 inches by 11 inches. When squares of side length x are cut from each corner, the new dimensions of the base of the box will be (8.5 - 2x) by (11 - 2x). The height of the box will be x.
2. <Step 2: Write the volume function.> The volume V of the box can be expressed as a function of x: V(x) = (8.5 - 2x)(11 - 2x)x.
3. <Step 3: Expand the volume function.> Expand the expression for V(x) to get a polynomial: V(x) = x(8.5 - 2x)(11 - 2x).
4. <Step 4: Use a graphing calculator.> Input the polynomial function V(x) into a graphing calculator. Use the table feature to evaluate V(x) for various values of x, ensuring x is within a reasonable range (0 < x < 4.25).
5. <Step 5: Identify the maximum volume.> Look for the maximum value of V(x) in the table. This value represents the maximum volume of the box, rounded to the nearest hundredth.>

Key Concepts

Volume of a Box

The volume of a box is calculated using the formula V = length × width × height. In this context, the dimensions of the box change as squares of side length x are cut from each corner of the paper. The new dimensions become (8.5 - 2x) for the length, (11 - 2x) for the width, and x for the height, leading to a polynomial expression for volume that can be maximized.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this problem, the volume of the box can be expressed as a polynomial in terms of x, which allows for the application of calculus or graphing techniques to find maximum values. Understanding how to manipulate and analyze polynomial functions is crucial for solving the problem.

Graphing Calculators and Tables

Graphing calculators are tools that can plot functions and create tables of values, which are essential for visualizing polynomial functions. By inputting the polynomial expression for volume into the calculator, students can generate a table of values to identify the maximum volume. Rounding to the nearest hundredth is a common practice in reporting results, ensuring precision in the final answer.