Problem 69 - College Algebra

Question

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 of the box is \(V(x) = 4x^3 - 32x^2 + 64x\).

Step 1: Understand the Shape of the Box

An open box is formed by cutting squares of side \(x\) inches from each corner of an 8-inch by 8-inch square cardboard. We then fold up the sides.

Step 2: Determine Remaining Dimensions

After removing squares of side \(x\), the new dimensions of the cardboard will be \((8 - 2x)\) inches for both the length and the width, since we remove \(x\) from two sides. The height of the box will be \(x\) inches.

Step 3: Write the Volume Formula

Volume \(V\) of a box is given by the formula \(V = \text{length} \times \text{width} \times \text{height}\). Here, it becomes: \(V(x) = (8 - 2x)(8 - 2x)x\).

Step 4: Simplify the Expression

First, calculate \((8 - 2x)(8 - 2x)\):\[(8 - 2x)^2 = 8^2 - 2\cdot8\cdot2x + (2x)^2 = 64 - 32x + 4x^2\]Now, multiply this result by \(x\):\[V(x) = x(64 - 32x + 4x^2) = 4x^3 - 32x^2 + 64x\]

Step 5: State the Final Polynomial Function

Thus, the volume of the box as a function of \(x\) is:\[V(x) = 4x^3 - 32x^2 + 64x\]

Key Concepts

Volume

Volume is a measure of the three-dimensional space that an object occupies. For our cardboard box, the volume represents how much space is inside the box once it's built.
Understanding volume is crucial because it helps us quantify the capacity of the box. In this case, we need to calculate the volume based on the dimensions we obtain after cutting and folding the cardboard.
The formula for the volume of an open box is the product of its length, width, and height. Here, the length and the width are the same because we're working with a square piece of cardboard, so:

• Volume = Length × Width × Height
• For our open box: \[V(x) = (8 - 2x)(8 - 2x)x\]

This formula will give us the volume as a function of the variable \(x\), which represents the side length of the squares cut out from each corner.

Open Box

An open box is a box without a lid, like the one described in our exercise. It starts as a flat cardboard sheet. When we cut out squares from each corner and fold the flaps upwards, it forms the sides.
This means our box only has four walls and a base.

• The height of this box comes from the small square corners we cut out. The side of the squares is \(x\), hence the height of the box is also \(x\).

The open top of the box allows for easy access to its contents, and making such boxes is a common practice in packaging. Understanding how the height relates to the cut-out squares helps in visualizing how changing \(x\) affects the final shape and volume of the box.

Cardboard Dimensions

The dimensions of the cardboard are crucial to forming the original box and understanding the transformation it undergoes.
Our starting cardboard piece is a square, specifically 8 inches by 8 inches. To create the box:

• Squares of side length \(x\) are removed from each corner.
• This changes the effective dimensions of the cardboard to \((8 - 2x)\) inches by \((8 - 2x)\) inches.

Folding along these new dimensions shapes the sides and bottom of the open box.
It's vital to recognize that as \(x\) increases, the unused portions of the cardboard form the box walls, which shrink the available dimensions of the base and, in turn, impact the volume.

Function Notation

Function notation is a way of representing mathematical relationships. With the exercise, we can express the relationship between \(x\) (the cut size) and the volume using function notation.
This allows us to easily study how changes in \(x\) affect the box's volume.

• Here, the volume is a function of \(x\), denoted as \(V(x)\).
• By using function notation, we state explicitly that volume depends on the cut size \(x\).

This is useful in various scenarios, such as predicting how larger or smaller cuts affect the actual space inside the box. Understanding and using function notation is essential for approaching similar problems where one variable influences another logically.