Problem 69 - College Algebra with Corequisite Support

Question

For the following exercises, use the written statements to construct a polynomial function that represents the required information. An open box is to be constructed by cutting out square corners of \(x\) - inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of \(x\)

Step-by-Step Solution

Answer

The volume function is \(V(x) = 4x^3 - 32x^2 + 64x\).

Step 1: Understand the Problem

We need to find the volume of a box formed by cutting out squares from each corner of an 8-inch by 8-inch square piece of cardboard and folding up the sides. The squares have sides of length \(x\).

Step 2: Calculate New Dimensions of the Box

After cutting out squares with side \(x\) from each corner, the length, width, and height of the box will change. The new length and width of the box will be \(8 - 2x\) (since squares are removed from both sides), and the height will be \(x\).

Step 3: Write the Volume Formula

The volume \(V\) of a rectangular box is given by the formula \(V = \, length \times width \times height\). Substitute the dimensions we found: \(V(x) = (8 - 2x)(8 - 2x)x\).

Step 4: Simplify the Volume Function

Expand \((8 - 2x)(8 - 2x)\) to get \(64 - 32x + 4x^2\). Substitute back to get \(V(x) = x(64 - 32x + 4x^2)\) and simplify to \(V(x) = 4x^3 - 32x^2 + 64x\).

Step 5: State the Final Volume Function

The polynomial representing the volume of the box as a function of \(x\) is \(V(x) = 4x^3 - 32x^2 + 64x\).

Key Concepts

Volume Formula

When dealing with three-dimensional objects like rectangular boxes, the volume formula is key to understanding how much space is contained inside. For a rectangular box, the volume can be calculated using the dimensions of length, width, and height. The formula is expressed as: \[ V = ext{length} \times ext{width} \times ext{height} \] Let's break it down:

• "Length" is the longest side of the box.
• "Width" is the side perpendicular to the length.
• "Height" corresponds to the depth of the box.

In our example, after cutting and folding the cardboard, the length and width become \(8 - 2x\), where \(x\) is the edge of the square cut-out, and the height is \(x\). This gives us a specific volume function for this scenario. By substituting these new dimensions into the volume formula, we find that the volume as a function of \(x\) is \(V(x) = (8 - 2x)(8 - 2x)x\). Understanding and applying this formula allows you to discover how the box's volume changes based on the size of the cut-out square.

Rectangular Box Dimensions

The dimensions of a rectangular box are crucial in determining both its volume and structure. When modifying dimensions, like cutting out corners from a piece of cardboard, it's important to adjust each dimension accurately. In our exercise, a square with side length \(x\) is removed from each corner of an 8-inch square piece of cardboard. After removing these corners, the dimensions of the cardboard change accordingly.

• The original length of 8 inches is reduced by the squares on both ends, resulting in a new length of \(8 - 2x\).
• The width similarly becomes \(8 - 2x\).
• Finally, the height, which corresponds to the sides being folded up, is equal to \(x\).

Thus, the new dimensions of the box transform the flat cardboard into a three-dimensional box, highlighting the dynamic relationship between size adjustments and the corresponding physical alterations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining their value. In mathematics, this often means combining like terms, distributing operations, and eliminating redundant elements. For our volume calculation, simplifying helps in uncovering a clear, streamlined polynomial function for the box. Starting with \(V(x) = (8 - 2x)(8 - 2x)x\), we need to simplify:

• First, expand \((8 - 2x)(8 - 2x)\). This results in \(64 - 32x + 4x^2\).
• Next, distribute \(x\) across these terms gives \(64x - 32x^2 + 4x^3\).
• Reordering by descending powers of \(x\), the final simplified expression is \(V(x) = 4x^3 - 32x^2 + 64x\).

These steps assist in crafting a clear polynomial that makes computations easier and quicker. Mastering simplification helps in managing complex expressions effectively, providing a solid math foundation.