Problem 75 - A Graphical Approach to College Algebra

Question

Volume of a Box \(\quad\) A standard piece of notebook paper measuring 8.5 inches by 11 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let \(x\) represent the length of a side of each such square in inches. (a) Use the table feature of your graphing calculator to find the maximum volume of the box. (b) Use the table feature to determine to the nearest hundredth when the volume of the box will be greater than 40 cubic inches.

Step-by-Step Solution

Answer

The maximum volume is approximately 61.26 cubic inches. The volume exceeds 40 cubic inches when \(x\) is between approximately 1.07 and 2.43 inches.

Step 1: Define the Volume Expression

The box is made by cutting squares of side length \(x\) from each corner of the 8.5 by 11 inch paper and folding up the sides. This results in a box with a length of \(11 - 2x\), a width of \(8.5 - 2x\), and a height of \(x\). The volume \(V(x)\) of the box can be expressed as \[V(x) = x(11 - 2x)(8.5 - 2x)\]

Step 2: Set Up the Table on the Calculator

Use your graphing calculator's table feature to evaluate the expression for various values of \(x\). Input the expression \(V(x) = x(11 - 2x)(8.5 - 2x)\) into the calculator. Choose a range of values for \(x\), starting from a small positive number up to a reasonable approximation where the volume potentially decreases to zero. Typically, you would try values between 0 and a little bit less than 4.25 (half of the shorter side, as the square cut can't exceed half of either side's length).

Step 3: Identify Maximum Volume

While reviewing the table, look for the value of \(x\) that yields the largest volume, \(V(x)\). This value of \(x\) corresponds to the maximum volume the box can achieve. Ensure to refine the values of \(x\) to an appropriate precision to find the exact maximum point shown in the table.

Step 4: Identify Volume Greater than 40 Cubic Inches

Continue looking through the table from Step 2 for the values of \(x\) where the volume \(V(x)\) is greater than 40 cubic inches. Record the range of \(x\) that satisfies \(V(x) > 40\). Adjust \(x\) in increments that allow you to pinpoint the nearest hundredth.

Key Concepts

Graphing Calculator

In algebra, a graphing calculator is a superb tool when approaching problems involving variables and equations. When you’re tasked with finding particular values or visualizing expressions, it becomes indispensable.
Use your graphing calculator to handle polynomial expressions, like the volume expression here, which results from cutting squares out of notebook paper. Try exploring different features like graph plotting and table setups to understand how the volume changes.

• Start by inputting algebraic expressions into your calculator.
• Create a table for specified ranges of input values to see different outputs.
• Refine your input values to get more precise results.

These capabilities aid in visualizing mathematical concepts, ensuring robust solutions are reached efficiently.

Maximum Volume

Determining the maximum volume of a box involves both understanding its dimensions and how the size of the cuts influences it. Here, finding the maximum volume means pinpointing the value of the variable that brings the volume of the box to its peak.
After defining the volume expression as a function of length, width, and height, import it into your graphing calculator. This expression is formulated initially as \[ V(x) = x(11 - 2x)(8.5 - 2x) \],where each dimension is adjusted by the cut.

• Use the table to evaluate the function over several increments of\( x \).
• Examine the table for the highest output value.
• Fine-tune your input to get the exact maximum volume value.

This step effectively leverages algebra and technology to solve optimization problems in geometry.

Volume Inequality

Volume inequalities require ensuring that a condition, such as the volume being greater than a certain amount, is satisfied. Using a volume expression, such as \[ V(x) = x(11 - 2x)(8.5 - 2x) \], determine when our box’s volume surpasses 40 cubic inches.
In a graphing calculator, this entails checking and evaluating the table for points where the volume first exceeds the desired quantity.

• Adjust the increments in your table for precision.
• Locate all intervals where \( V(x) > 40 \).
• Document these values to determine the probable range for your solution.

This approach applies inequalities practically, demonstrating how constraints are addressed in real-world applications like optimizing container dimensions.

Algebraic Expression

Understanding and composing the algebraic expression that defines a problem is crucial when calculating dimensions like volume. An algebraic expression such as \[ V(x) = x(11 - 2x)(8.5 - 2x) \], is formed to reflect the relationship between variables and the problem's constraints.
This expression can be broken down into a factor involving:

• \( x \) representing the height of the box.
• \( 11 - 2x \) for the adjusted length.
• \( 8.5 - 2x \) being the new width.

By solving this equation, you reveal insights about how these parameters affect one another. Understandably, creating and solving such expressions enables insights into the geometry and optimization challenges we face.