Introduction

In the realm of geometry and practical application, determining the optimal dimensions of a box from a rectangular cardboard sheet is a common problem. Specifically, given a cardboard sheet with dimensions of 30 cm x 50 cm, and the need to cut four squares from the corners, the challenge is to find the dimensions of these squares such that the remaining part, when folded, yields a box with the largest possible surface area. This article delves into the mathematical derivation and provides a clear understanding of the process to achieve this goal.

Mathematical Derivation

Let each of the four squares cut from the corners be of dimension ('n' cm by 'n' cm. When these squares are removed and the remaining part of the sheet is folded along the edges, a box with dimensions (50 - 2n) cm by (30 - 2n) cm by (n) cm is formed.

The total surface area ((s)) of the box can be expressed as:

[ s 2 times ((50 - 2n) times (30 - 2n)) 2 times (50 - 2n) times n 2 times (30 - 2n) times n ]

Expanding and simplifying this expression gives:

[ s 2(1500 - 160n 4n^2) 100n 60n ]

Further simplification:

[ s 3000 - 320n 8n^2 ]

This expression can be rewritten as:

[ s 8n^2 - 320n 1500 ]

The goal is to find the value of (n) that maximizes (s). To do this, we will find the vertex of the parabola represented by this quadratic equation, as the vertex will give the maximum value since the coefficient of (n^2) is positive (indicating an upward-opening parabola).

The vertex form of a quadratic equation (ax^2 bx c) is given by ( n -frac{b}{2a} ). Here, (a 8) and (b -320).

[ n -frac{-320}{2 times 8} frac{320}{16} 20 div 2 10 text{ cm} ]

Therefore, the dimensions of the squares to be cut from the corners are 10 cm by 10 cm.

Substituting (n 10) back into the dimensions of the box:

[ text{Length} 50 - 2 times 10 30 text{ cm} ] [ text{Width} 30 - 2 times 10 10 text{ cm} ] [ text{Height} 10 text{ cm} ]

With these dimensions, the total surface area of the box is maximized.

Discussion

The problem highlights the application of quadratic optimization in practical scenarios. By cutting the squares from the corners and folding the sheet, the dimensions of the resulting box can be precisely controlled to achieve the maximum surface area.

The analysis reveals a critical point where the surface area is maximized, providing insights into how small changes in the dimension of the cut squares ((n)) significantly affect the final shape and size of the box.

Conclusion

In conclusion, when cutting four squares of 10 cm by 10 cm from a 30 cm x 50 cm rectangular cardboard sheet, and folding the remaining part, the resulting box will have a surface area of maximum possible size. This problem showcases not only the mathematical principles of optimization but also their practical application in real-world scenarios.

Keywords: cardboard box, surface area, optimization, geometry