Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

From a rectangular piece of cardboard having dimensions 20 inches \(\times 30\) inches, an open box is to be made by cutting out an identical square of area \(x^{2}\) from each corner and turning up the sides (see the figure). Express the volume \(V\) of the box as a function of \(x\).

Step-by-Step Solution

Answer

The volume \(V\) of the box is expressed as a function of \(x\): \(V(x) = 4x^3 - 100x^2 + 600x\).

Step 1: Understand the problem

We begin with a rectangular piece of cardboard measuring 20 inches by 30 inches. We need to cut out equal squares of side length \(x\) from each corner to form an open box by folding up the sides.

Step 2: Dimensions after cutting

After cutting out squares of side length \(x\) from each corner, the new length and width of the cardboard become \((30 - 2x)\) and \((20 - 2x)\) respectively, as \(x\) is cut out from both sides of the length and width.

Step 3: Determine the height of the box

The height of the box after folding up the sides will be equal to \(x\), as each corner square's side length is \(x\), which is what defines the height when folded.

Step 4: Write the volume formula

The volume \(V\) of an open box is given by the formula \(V = \text{length} \times \text{width} \times \text{height}\). Substituting the dimensions, the volume becomes \(V = (30 - 2x)(20 - 2x)(x)\).

Step 5: Simplify the volume expression

First, multiply the expressions for the new length and width: \((30 - 2x)(20 - 2x) = 600 - 40x - 60x + 4x^2 = 600 - 100x + 4x^2\). Then multiply by \(x\) for the volume: \(V = x(600 - 100x + 4x^2) = 4x^3 - 100x^2 + 600x\).

Key Concepts

Rectangular Cardboard

The concept of a rectangular cardboard relates to the initial shape and size of the material you're working with. In this specific exercise, you start with a cardboard piece measuring 20 inches by 30 inches. This size is crucial because it influences all subsequent steps involved in constructing the box.
The cardboard serves as the raw material from which the box will be crafted. It's important to visualize it as a flat surface before any modifications. This rectangular shape not only defines the available material but also determines how effectively you can maximize the box's volume. Understanding the dimensions of the cardboard is the first step towards visualizing the transformation process, from plan to product without any initial cuts.

Box Dimensions

Once you decide to construct an open box, the original dimensions of the cardboard undergo alterations. Cutting a square out from each corner will change both the length and width of the cardboard.
Specifically, when you cut out a square with a side length of \(x\) from each corner, the effective length of the cardboard becomes \(30 - 2x\) inches, and the width becomes \(20 - 2x\) inches. This is because the cuts on the corners remove \(x\) inches from each end of the length and width.

• Initial Length: 30 inches
• Initial Width: 20 inches
• Adjusted Length: \(30 - 2x\) inches
• Adjusted Width: \(20 - 2x\) inches

Understanding how each cut impacts the box's dimensions helps in planning the construction and anticipating the end volume you can achieve. Defining these dimensions clearly is essential for accurate calculation in further steps.

Volume Function

Creating a volume function involves determining the three dimensions of the open box and then using these to write an expression that calculates the box's volume. This exercise requires understanding how the box's volume changes based on the variable \(x\), which is the side length of the removed squares.
The dimensions of the box, after cutting and folding, are:

• New Length: \(30 - 2x\)
• New Width: \(20 - 2x\)
• Height (equal to square side): \(x\)

The volume function can then be expressed using the formula: \[V = ext{length} \times ext{width} \times ext{height} = (30 - 2x)(20 - 2x)x\] After expanding and simplifying, the volume function becomes: \[ V = 4x^3 - 100x^2 + 600x \] This cubic polynomial shows how the volume depends on the different values of \(x\), providing insight into maximizing the box's volume during construction.

Open Box Construction

The construction of an open box from a rectangular cardboard involves strategic cutting, folding, and assembling. The idea is to use the cardboard efficiently to create a box structure without a top.
Once you've cut a square of side length \(x\) from each corner, you then fold up the sides. This folding process turns the remaining flaps into the sides of the box, with each original square cut providing a side's height.

• The height of the box is determined by the square's side length \(x\), which also increases the structural integrity of the box once folded.
• Understanding the folding pattern and dimensions left behind after the cut is crucial for a sturdy construction.

Visualizing the transition from a flat rectangular shape to a three-dimensional open box helps students grasp the practical aspect of volume and dimensions.