Question

You need to wrap a rectangular box with dimensions 2 feet by 3.5 feet by 4.2 feet. What is the least amount of wrapping paper you must buy in order to complete the job?

Short answer

60.2 square feet of wrapping paper is needed.

Step-by-step solution

Understand the Problem

To determine the least amount of wrapping paper needed, we must find the surface area of the rectangular box. The surface area is essentially the total area of all the sides of the box that need to be covered.

Identify the Dimensions

The dimensions of the box are given as 2 feet (length), 3.5 feet (width), and 4.2 feet (height).

Recall Surface Area Formula for a Rectangular Box

The formula for the surface area of a rectangular box is given by:\[SA = 2(lw + lh + wh)\]where \(l\), \(w\), and \(h\) are the length, width, and height of the box respectively.

Substitute the Values into the Formula

Substitute the given dimensions into the surface area formula:\[SA = 2((2 \times 3.5) + (2 \times 4.2) + (3.5 \times 4.2))\]

Calculate Each Term

First calculate each term individually:- \(2 \times 3.5 = 7\)- \(2 \times 4.2 = 8.4\)- \(3.5 \times 4.2 = 14.7\)

Sum the Terms Inside the Parentheses

Add the calculated terms:\[7 + 8.4 + 14.7 = 30.1\]

Calculate the Total Surface Area

Multiply the sum by 2 to find the total surface area:\[SA = 2 \times 30.1 = 60.2 \]

Conclude the Least Amount of Paper Needed

The least amount of wrapping paper needed is 60.2 square feet.

Key concepts

Rectangular Box

A rectangular box is a three-dimensional shape, resembling what most commonly refers to as a 'box'. It has six flat faces, each of which is a rectangle. A key aspect of a rectangular box is that opposite faces are identical in size. This basic geometry concept is used in many real-world applications, such as packaging.
Understanding a rectangular box involves knowing its dimensions: length, width, and height. These parameters allow us to calculate more complex properties, such as the surface area or volume. The surface area is the sum of the areas of all the faces of the box. In the context of wrapping a box, like in the original exercise, the surface area gives us the necessary information to determine how much wrapping paper is needed.

• Length (l): One of the longer sides of the rectangle.
• Width (w): The shorter side of the base rectangle.
• Height (h): The vertical side, perpendicular to the base.

Each dimension plays a crucial role in mathematical formulas and is integral to solving geometric problems like the one we are addressing here.

Geometry Problem Solving

Geometry problem solving often requires a systematic approach, which is critical when calculating attributes such as the surface area of a rectangular box. The step-by-step method used in the solution is an excellent roadmap for tackling these types of problems. It starts with understanding the problem and identifying what needs to be solved.
Once we comprehend the problem, as seen in steps 1 and 2 of the original solution, we can identify the necessary dimensions of the box. Utilizing the formula for the surface area of a rectangular box, which accounts for all the faces of the box, is the next step. Knowing the formula:

• Formula: \(SA = 2(lw + lh + wh)\)
• Components: This includes calculating the area for each pair of opposite faces.

In essence, the equation's beauty lies in its simplicity—there are three separate products you need for one half of the box, and you multiply by two to include both halves.

Mathematics Education

In the realm of mathematics education, problem-solving exercises like these play an integral role in developing analytical skills. They teach students how to break down complex tasks into manageable steps. This approach not only builds confidence but also leads to mastery of concepts.
Effective mathematics education involves teaching students how to:

• Apply formulas correctly.
• Understand the relationship between different geometric properties.
• Develop reasoning skills that can be applied across various fields.

The exercise of wrapping a rectangular box exemplifies practical applications of mathematics in everyday life, instilling a sense of relevance and importance. Moreover, by engaging in such exercises, students learn how geometry is not just a subject but a useful tool, applicable in numerous areas.