Problem 37 - Thomas Calculus

Question

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side \(x\) at each corner and then folding up the sides as in the figure. Express the volume \(V\) of the box as a function of \(x .\)

Step-by-Step Solution

Answer

The volume function is \(V(x) = (22 - 2x)(14 - 2x)x\).

Step 1: Understanding the Problem

We have a rectangular piece of cardboard with dimensions 14 in. by 22 in. and we need to cut out squares of side length \(x\) from each corner to create a box with an open top. Our goal is to express the volume of this box, \(V(x)\), as a function of \(x\).

Step 2: Identify Dimensions After Cutting Squares

After cutting out squares of side \(x\) from each corner, the remaining length of the cardboard will be \(22 - 2x\) (since squares are cut from both sides of the length) and the remaining width will be \(14 - 2x\) (similarly for the width).

Step 3: Determine Height of the Box

After folding up the sides of the box, the height of the box will be equal to the side length of the square cut out, which is \(x\).

Step 4: Express Volume as a Function of x

The volume \(V\) of the box is expressed as the product of its length, width, and height. Therefore, the volume can be expressed as:\[ V(x) = (22 - 2x)(14 - 2x)x \]This equation represents the volume \(V\) as a function of the side length \(x\) of the squares cut out.

Key Concepts

Rectangular Box

A rectangular box is a three-dimensional shape with six flat faces, including a length, width, and height. Each face is a rectangle. The concept of a rectangular box is fundamental in geometry and everyday applications, like packaging and storage design. In our exercise, the rectangular box is formed by a piece of cardboard with specific dimensions: 14 inches by 22 inches. By making cuts and folds, a rectangular box with only five visible faces is created, as one face is open unlike traditional closed rectangular boxes.

Open-Top Box

An open-top box is essentially like a regular box but without a cover on one side. This characteristic is important in practical uses where easy access or ventilation is needed. In terms of construction from a rectangular piece of cardboard, you create an open-top box by cutting squares from the corners and folding the flaps upwards. This particular process of forming a box without a lid plays a significant role, especially in the packaging industry, where materials and cost savings are essential. By removing the top, manufacturers can conserve materials and still provide sufficient containment.

Volume Function

The volume function of a geometric shape provides a formula to calculate how much space the shape occupies in three dimensions. Volume is crucial when trying to determine the capacity of a box. For the open-top box in this exercise, the volume function is derived by identifying the internal dimensions of the box, which change when squares with side length \( x \) are removed.
The volume function \( V(x) \) is given by multiplying the internal length, width, and height of the box: \( V(x) = (22 - 2x)(14 - 2x)x \).
Here, \( 22 - 2x \) and \( 14 - 2x \) denote the adjusted length and width after the corners are removed, and \( x \) is the height, represented by the sides that are folded upwards. Calculating the volume using this function allows us to determine how changing \( x \) affects the overall space available in the box.

Cardboard Construction

Constructing a box from a flat piece of cardboard involves a series of strategic cuts and folds. In the case of an open-top rectangular box, you start with a flat rectangular sheet. By cutting equal-sized squares from each corner of the cardboard and folding the resultant flaps upwards, you craft the structural sides of the box.
These steps involve:

• Identifying the size of squares to cut that will yield an optimal box size for specific needs.
• Understanding how to maximize the dimensions of the box's height, which is governed by the size of the squares \( x \) cut out.

Effective cardboard construction ensures material efficiency and the desired box volume, making it a vital skill in packaging and manufacturing industries.