Extended Essay (Mathematics)

International Baccalaureate Diploma 2010

Abstract

My research question in this essay is: How can the AM-GM inequality be applied in finding bounds on the areas and volumes of two-dimensional and three-dimensional geometric shapes respectively? I begin by introducing the inequality and briefly, its practical applications. Then, I prove the inequality using conventional and a more unique type of mathematical induction, as well as determining the conditions for which the equality happens. Afterwards, I show the geometric interpretation of the inequality and begin my exploration of two-dimensional shapes with the triangle. I attempt to find a relationship between equilateral, isosceles and scalene triangles using the AM-GM inequality and which one gives the maximum area when all these triangles have the same perimeter. I prove that a regular triangle has the greatest area given the same perimeter. Then, I explore shapes with a greater number of sides using a similar approach. My final generalization is that the regular polygon gives the greatest area given same perimeter. I extend the project by exploring circular shapes and coming up with unique properties that differentiate circular shapes from the rest. Also, I use the AM-GM inequality to determine other inequalities of a shape; this is actually meant to further show the various techniques in applying the AM-GM inequality. My three-dimensional shape exploration involves looking at different shapes, as three-dimension offers room for more variety of shapes. My final generalization is that no fixed dimension gives all three-dimensional shapes maximum volume; all have their unique properties. I have actually done a study on this inequality in my earlier school years. However, I felt that I did not explore deeply enough the inequality’s geometric applications. I re-used some of the problems I analyzed in my previous study, but my method of exploration is almost entirely different. 294 words

Table of Contents

1. IntroductionPg 4

2. Proof of AM-GM Inequality by Mathematical Induction Pg 5-8

3. a) Two-dimensional geometric interpretation of the AM-GM inequality Pg 9 b) Two-dimensional geometric problems involving the AM-GM Inequality Pg 10-22

4. Three-dimensional geometric problems involving the AM-GM Inequality Pg 23-27

5. Conclusion Pg 28

6. Bibliography Pg 29-30

7. Appendix Pg 31-34

Introduction

The inequality of arithmetic and geometric means (AM-GM inequality) states that the arithmetic mean (average) of a set of non-negative real numbers is greater or equal to the geometric mean. The two means are equal if and only if the set of numbers in the list are all the same.

x1+x2+…+xnn≥nx1x2…xn

and x1+x2+…+xnn=nx1x2…xn

if and only if x1=x2=…=xn

The AM-GM inequality can be applied in various geometrical problems. It is useful especially in determining maximum and minimum quantities in geometry on paper and actual three-dimensional shapes. Therefore we can see that the AM-GM inequality is not only applicable in solving geometrical problems in mathematics competitions such as Mathematical Olympiads, but it has real-life applications as well. Take for example this simple scenario: a storage company has a certain amount of thin wood used to build boxes, and of course it would want to maximize the volume of the boxes so that it can store the most material. The AM-GM inequality can be directly applied to determine the solution to maximizing volume. Thus, the importance of the AM-GM inequality in geometry is established. What also got me interested in the AM-GM inequality was when I realized it is an alternative means to calculus in obtaining maximum and minimum values in graphs (see...