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. Determine when the volume of the box will be greater than 40 in.^3.

Verified step by step guidance

1. Step 1: Define the volume function. The volume V of the box can be expressed as a function of x, where x is the side length of the square cut from each corner. The dimensions of the box will be (8.5 - 2x) by (11 - 2x) by x. Therefore, the volume function is V(x) = x(8.5 - 2x)(11 - 2x).
2. Step 2: Set up the inequality. We want to find the values of x for which the volume is greater than 40 cubic inches. This means we need to solve the inequality V(x) > 40, or x(8.5 - 2x)(11 - 2x) > 40.
3. Step 3: Use a graphing calculator. Enter the volume function V(x) = x(8.5 - 2x)(11 - 2x) into the graphing calculator. Use the table feature to find values of x that make V(x) greater than 40.
4. Step 4: Analyze the table. Look for the range of x values in the table where the volume is greater than 40. Pay attention to the intervals and note the smallest and largest x values that satisfy the inequality.
5. Step 5: Round the results. Once you have identified the range of x values, round these values to the nearest hundredth to find the precise range of x for which the volume of the box is greater than 40 cubic inches.

Key Concepts

Volume of a Box

The volume of a box is calculated using the formula V = length × width × height. In this scenario, the box is formed by cutting squares from the corners of a rectangular sheet, which affects its dimensions. The new dimensions after cutting squares of side length x will be (8.5 - 2x) for the length, (11 - 2x) for the width, and x for the height. Understanding how these dimensions interact is crucial for determining the volume.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is greater than, less than, or equal to the other. In this problem, we need to set up an inequality to find when the volume of the box exceeds 40 in³. This involves solving the inequality V > 40, which will help identify the permissible values of x that satisfy this condition.

Graphing Calculators and Tables

Graphing calculators are powerful tools that can visualize mathematical functions and inequalities. By using the table feature, one can input the volume function derived from the box's dimensions and observe the corresponding values of x. This allows for a systematic approach to find when the volume exceeds 40 in³, as it provides a clear view of how the volume changes with different values of x.