APPLICATION EXAMPLES TO PROBLEMS OF MODERN MATHEMATICAL APPARATUS

Abstract

Purpose. The purpose of the research work is: – possibility to combine different ways of solving certain mathematical problems. In general, the term “non-standard” methods of solving problems in mathematics has not defined yet, but many authors use this term in their researchers. It should be noted that there are many school problems that use unusual considerations. These are the tasks that are considered to be more complex and require non-standard methods of solving. These methods illustrate the wide possibilities of using well-acquired school knowledge and instill skills in using non-standard methods of reasoning in solving problems; – the performance of comparative analysis in the calculations of spent work with different adjacent geometry of three-dimensional figures, which are given in this work; – establishing a mathematical law for calculating the maximum number of embeddings of a set of homogeneous circular objects inside a certain external geometric structure. Methodology. Research of this work is based on the use of modern mathematics, such as school and analytical geometry, the basics of integral calculus and their practical application, progression. Practical implications. The first part of the paper presents several different methods for solving one geometric (stereometric) problem using both elementary geometry and higher mathematics, in particular, analytical geometry. These different ways of solving one specific problem demonstrate the versatility of the modern mathematical apparatus, link the mutual goals and methods of elementary and higher mathematics in specific applications. It is shown that the problem with the school formulation of the condition can be solved by means of higher mathematics with the use of actions on vectors, the use of types of products of vectors and so on. The second part considers mathematical models of a specific technical problem based on the known law of physics, such as the calculation of some work under the action of gravity in a particular case. As such work, as it is established, is differentiated on a certain independent variable, therefore at first the value of a separate element of such work is established and integration of this element on this variable within its limits is executed. The given models with different geometry of the location of the three-dimensional body (reservoir, etc.), for which the study of this numerical characteristic is studied, are studied and compared, a comparative analysis is made in these two different positions of this body. The third part investigates the maximum filling of certain geometric both flat and three-dimensional external structures with many circular (spherical) figures, establishes a mathematical law for calculating the quantitative characteristics of such maximum filling, proposed and tested the criterion (coefficient) of efficiency and usefulness of such filling. For each of the parts of the given researches the corresponding figures, tables which supplement accordingly received results in the form of formulas and calculations are offered. Conclusions have been also made on the research conducted in the work. Value/originality. The originality of the research is as follows: – use and combination of school and higher mathematics in solving a specific geometric problem; – the use of mathematical apparatus in calculating the work of solving a physical problem; – establishment of the mathematical law of the maximum filling of a certain geometric structure by a homogeneous set of circular figures. By solving one geometric problem in different methods or ways, it is possible to better understand the specific method, its advantages and disadvantages depending on the content of the problem. The use of different methods of such a solution provides an opportunity to replace it with another solution, which encourages to find alternative effective creative approaches to solving this problem. It is not necessary to solve each problem in different ways or methods, just to choose one or two. In order to enhance cognitive activity and learn different methods of solving geometric problems, it is proposed to use nonstandard methods of solving geometric problems. Regarding the amount of work spent, the main element of novelty is the comparative analysis of such calculations in two related cases of the location of a geometric body (cylinder): when its foundations are vertical or horizontal. The main element of the novelty of the results of the maximum filling of one geometric structure by a set of circular objects is the establishment of the mathematical law of quantitative characteristics of such filling, while proposing and testing a logical convenient coefficient of usefulness of such maximum filling.