THE ROLE OF MATHEMATICAL CONCEPTS IN REAL-WORLD PROBLEMS

ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 57 THE ROLE OF MATHEMATICAL CONCEPTS IN REAL-WORLD PROBLEMS Bekchanova Shakhnoza Khusanboy qizi (masters degree holder in Mathematics, Urgench State University named after Abu Rayhan Beruni, Email: [email protected], Telephone: +998937582508) Annotation: This article shows how mathematics applies to real-life problems. It includes designing a sustainable box using calculus, solving the Stable Matching Problem with algorithms, and finding a dam’s area using integrals, highlighting math’s role in solving practical challenges. Key words: optimization, stable matching, algorithms, integral, geometry, sustainability. Introduction. This paper presents three real-world applications of mathematics that demonstrate how abstract concepts can solve practical problems. The first case applies calculus to design a sustainable package using minimal material. The second explores computational thinking through the Stable Matching Problem, showing how algorithms can ensure fairness in assignments. The third uses integral calculus to determine the area of a gravity dam’s cross-section, combining geometry and integration. Together, these examples highlight the versatility of mathematical tools in optimizing design, supporting decisionmaking, and solving engineering challenges efficiently. Rational function for sustainable package: a case study (Calculus) Imagine a packaging company needs to design a shipping box with a square base and a volume of 600 cubic inches. What dimensions should it have? Should the base be 5 5 inches, 10 10 inches, or something else? Which one uses the least material? ( 1 inch = 2.54 cm) Solution: We know the volume of the box must be 600 cubic inches, so: 2 600 l h= Now, substitute this into the surface area formula: 2 2 2 600 2400 4 2 2S l l l l l = + = + ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 58 Let's now find the minimum surface area using calculus by taking the derivative of the surface area function: ( ) 22400 2S l l l = + 2 2 2400 2400 2 4 dS d l l dl dl l l = + = - . To find critical points (possible minimums), we will solve: 2 2400 4 0ll - = 3600 8.43l= = Second derivative test to check if it is a minimum 2 2 2 3 2400 4800 4 4 d S d l dl dl l l = - = + Since this is always positive for 0l> , the function has a minimum at 8.43l= . From this we can always find 8.43h= . This means the box is a cube with all sides approximately 8.43 inches, and it uses the least material for a fixed volume of 600 cubic inches. The Stable Matching Problem (Computational Thinking) Every year, thousands of students graduate from medical schools and begin their careers by applying for residency training positions in hospitals. Each medical student has a personal ranking of the hospitals where they would prefer to work, and each hospital has its own ranking of students based on academic merit, interviews, and institutional fit. Students: John, Alice, Maria, Ethan, Noor. Hospitals: Hospital A, Hospital B, Hospital C, Hospital D, Hospital E. Student 1st 2nd 3rd 4th 5th John B A D E C Alice A C B D E Maria B D E A C ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 59 Ethan C B A D E Noor D E C B A Hospital 1st 2nd 3rd 4th 5th A Ethan Alice John Maria Noor B John Maria Alice Noor Ethan C Ethan Noor Alice John Maria D Noor Maria Ethan Alice John E Alice Ethan Noor Maria John Solution. Let’s solve using Gale–Shapley. Step 1: Everyone proposes to 1st choice John → B Alice → A Maria → B Ethan → C Noor → D Hospitals evaluate:B:John, Maria → prefers John (1st), rejects Maria A:Alice → accepts C:Ethan → accepts D:Noor → accepts Step 2: Maria proposes to 2nd choice D already holds Noor. D ranks: Noor (1st), Maria (2nd) → keeps Noor, rejects Maria again. Step 3: Maria proposes to 3nd choice: E is free, accepts Maria. Final matching: Student Hospital John B Alice A Maria E ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 60 Ethan D Noor C Area of the Gravity Dam (Calculus) Image 1. You can see that the gravity dam's cross-section is composed of three fundamental geometric shapes: a rectangle, a right triangle, an area under the parabola. Image 2. 2 389 16 6224 A Area h b ft= = = 2 1 1 389 59 11475 2 2 B Area h a ft= = = We can break it into these three pieces. We can use simple geometry to calculate the area of the rectangular piece and the triangular piece, which leaves us with calculating the area of the parabolic piece. This is where the calculus will be used. ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 61 Image 3. We are using integral calculus to calculate this. If we keep making the rectangles smaller and smaller, we will eventually get the exact area under the curve. The curve starts at the origin ( ) 0,0 and rises to a height of 389 ft at the end of the 54 ft wide base. To model the curve, we use a simple quadratic function: 2 ( ) C f x ax= Using the known point ( ) 54,389 , we find the value of a : 389 2916 a= . So the function becomes: 2 389 ( ) 2916 C f x x= . Now we will use Riemann integral: ( ) ( ) 0 lim i b C C i i C xa i Area f x x f x dx ® = = VV ( ) 3 54 54 2 2 0 0 389 389 54 5803 2916 2916 3 C C Area f x dx x dx ft= = = » 2 23502TotalArea ft= . ISSN: 3030-3931, Impact factor: 7,241 Volume 10, issue 2, Noyabr 2025 https://worldlyjournals.com/index.php/Yangiizlanuvchi worldly knowledge OAK Index bazalari : research gate, research bib. Qo’shimcha index bazalari: zenodo, open aire. google scholar. Original article 62 References: 1. Kehle, P., & Carpenter, T. (2016). Fair and Stable Matchings: Computational Thinking Module (Student Edition). COMAP, Inc., in conjunction with DIMACS, Rutgers University. 2. Rational Function for Sustainable Package: A Case Study. Retrieved from https://dartef.com/blog/functions-sustainability-1/ 3. AP Calculus BC Project: The Dam Problem – Constructing a Dam across the River. Retrieved from https://www.ms.uky.edu/ 4. Dams. Retrieved from https://damsafety.org/