The Total Area of the Cutouts

Oct 13, 2025 by miodeya.com 118 views

Introduction

In the world of mathematics, problems often arise from everyday situations. One such scenario is when Samson attempts to create a shipping box out of a 2-foot by 4-foot piece of cardboard. His plan involves cutting two squares of side length $x$ and two rectangles with a width of $x$ out of the cardboard, and then folding the cardboard along the lines to make a shipping box. In this article, we will delve into the mathematical aspects of Samson's project and explore the constraints and possibilities of his design.

The Cardboard's Dimensions

The cardboard is initially 2 feet by 4 feet, which can be expressed as a rectangle with dimensions $l = 2$ and $w = 4$ . To create the shipping box, Samson will cut two squares of side length $x$ and two rectangles with a width of $x$ . The total area of the cardboard is given by the formula $A = l \times w$ , which in this case is $A = 2 \times 4 = 8$ square feet.

Cutting the Squares

The two squares of side length $x$ will have an area of $x^2$ each. Since there are two squares, the total area of the squares is $2x^2$ . This means that the remaining area of the cardboard after cutting the squares will be $8 - 2x^2$ .

Cutting the Rectangles

The two rectangles with a width of $x$ will have a length of $l - 2x$ , as the length of the cardboard is $l = 2$ and the width of the rectangles is $x$ . The area of each rectangle is given by the formula $A = l \times w$ , which in this case is $A = (2 - 2x) \times x = 2x - 2x^2$ . Since there are two rectangles, the total area of the rectangles is $2(2x - 2x^2) = 4x - 4x^2$ .

The Total Area of the Cutouts

The total area of the cutouts is the sum of the areas of the squares and the rectangles, which is $2x^2 + 4x - 4x^2$ . Simplifying this expression, we get $4x - 2x^2$ .

The Constraints

The constraints of Samson's design are that the total area of the cutouts must be less than or equal to the total area of the cardboard, which is $8$ square feet. This can be expressed as the inequality $4x - 2x^2 \leq 8$ . Additionally, the side length of the squares must be less than or equal to the width of the cardboard, which is $4$ feet. This can be expressed as the inequality $x \leq 4$ .

The Possibilities

To find the possible values of $x$ , we need to solve the inequality $4x - 2x^2 \leq 8$ . This can be rewritten as $2x^2 - 4x + 8 \geq 0$ . Factoring the left-hand side, we get $(x - 2)^2 \geq 0$ . Since the square of any real number is always non-negative, this inequality is true for all values of $x$ . Therefore, the possible values of $x$ are all real numbers greater than or equal to $0$ .

The Optimal Value of $x$

To find the optimal value of $x$ , we need to maximize the area of the shipping box. The area of the shipping box is given by the formula $A = l \times w$ , where $l$ is the length of the box and $w$ is the width of the box. Since the box is made by folding the cardboard along the lines, the length of the box is $l = 2 - 2x$ and the width of the box is $w = 4 - 2x$ . The area of the box is therefore $A = (2 - 2x)(4 - 2x) = 8 - 12x + 4x^2$ .

To maximize the area of the box, we need to find the value of $x$ that maximizes the expression $8 - 12x + 4x^2$ . This can be done by taking the derivative of the expression with respect to $x$ and setting it equal to $0$ . The derivative of the expression is $-12 + 8x$ . Setting this equal to $0$ , we get $-12 + 8x = 0$ , which implies that $x = \frac{3}{2}$ .

Conclusion

Q: What is the main goal of Samson's project?

A: The main goal of Samson's project is to create a shipping box out of a 2-foot by 4-foot piece of cardboard by cutting two squares of side length $x$ and two rectangles with a width of $x$ out of the cardboard, and then folding the cardboard along the lines to make a shipping box.

Q: What are the constraints of Samson's design?

A: The constraints of Samson's design are that the total area of the cutouts must be less than or equal to the total area of the cardboard, which is $8$ square feet, and the side length of the squares must be less than or equal to the width of the cardboard, which is $4$ feet.

Q: What is the possible range of values for $x$ ?

A: The possible range of values for $x$ is all real numbers greater than or equal to $0$ .

Q: What is the optimal value of $x$ ?

A: The optimal value of $x$ is $\frac{3}{2}$ .

Q: How does the area of the shipping box change with respect to $x$ ?

A: The area of the shipping box is given by the formula $A = (2 - 2x)(4 - 2x) = 8 - 12x + 4x^2$ . As $x$ increases, the area of the box decreases.

Q: What is the maximum area of the shipping box?

A: The maximum area of the shipping box occurs when $x = \frac{3}{2}$ , and the maximum area is $A = (2 - 2 \times \frac{3}{2})(4 - 2 \times \frac{3}{2}) = 8 - 12 \times \frac{3}{2} + 4 \times (\frac{3}{2})^2 = 8 - 18 + 9 = -1$ .

Q: Is it possible to create a shipping box with a positive area?

A: Yes, it is possible to create a shipping box with a positive area. However, the area of the box will be less than the maximum area of $-1$ square feet.

Q: What are some potential applications of Samson's design?

A: Samson's design could be used to create custom shipping boxes for small items, such as jewelry or electronics. The design could also be used to create boxes for larger items, such as furniture or appliances.

Q: What are some potential limitations of Samson's design?

A: One potential limitation of Samson's design is that it may not be suitable for large or heavy items, as the cardboard may not be able to support the weight. Additionally, the design may not be suitable for items that require a lot of protection, such as fragile electronics.

Q: Can Samson's design be modified to improve its performance?

A: Yes, Samson's design can be modified to improve its performance. For example, the design could be modified to use a stronger material, such as cardboard with a higher density, or to add additional support structures, such as internal frames or reinforcement.

Q: What are some potential future directions for research on Samson's design?

A: Some potential future directions for research on Samson's design include:

Investigating the use of different materials, such as paper or plastic, to create the shipping box.
Developing new methods for cutting and folding the cardboard to create the shipping box.
Investigating the use of additional support structures, such as internal frames or reinforcement, to improve the performance of the shipping box.
Developing new algorithms and mathematical models to optimize the design of the shipping box.