AN EXTENSION OF THE ISOPERIMETRIC PROBLEM TO CERTAIN THREE-DIMENSIONAL OBJECTS

• Main Author: Setiawan, Andri
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/74583
• Institution: Institut Teknologi Bandung

In this thesis, we explore several extensions of the isoperimetric problem to three-dimensional figures, with a focus on right prisms and oblique prisms with rectangular, triangular, regular hexagonal, and circle bases. The isoperimetric problem for three-dimensional figures involves finding the shape or configuration that maximizes or minimizes a measure (such as volume) restricted by another measure (such as surface area). The main problem addressed in this paper is to find the prism shape that gives the largest volume while keeping the size of surface area. Through algebraic manipulation and simple trigonometry in the isoperimetric inequality, we obtain proof that a right prism volume’s will be larger than an oblique prism if their surface area and base are equal. Since the base area for both prisms is the same, we can compare their heights to determine which prism has a larger volume. The Cavalieri's Principle is used to determine the volume of the right prism and oblique prism. As the surface area and base of both prisms are the same, the height of the oblique prism (d) can be substituted with an equation involving the height of the right prism (t), resulting in an equation for the volume of the oblique prism that includes the volume of the oblique prism multiplied by a factor. The proof that this multiplier is less than one is key to demonstrating that the right prism will give a larger volume than the oblique prism if their surface area and base are the same. By applying the concept of extrema of functions of two variables, we conducted an analysis to determine the side lengths that result the maximum volume for right prisms with rectangular, triangular, regular hexagonal, and circle bases, where the surface area is given. A study was also conducted on the second derivative of the volume function to determine the type of critical points obtained. The results show that for right prisms, a regular base shape will give the largest volume. Another finding from this analysis is presented in a conjecture related to the base area that will give the largest volume for prisms. An examination of the extrema of the volume function for oblique prisms (one lateral side) with a rectangular base shows that a non-square base configuration leads to the maximum volume. The revised conclusion regarding the side lengths in the isoperimetric problem for prisms with rectangular bases is that a right prism with a square base will be larger than an oblique prism with a rectangular base with a certain configuration, and an oblique prism with a rectangular base with a certain configuration will be larger than an oblique prism with a square base. The results obtained can serve as a basis for investigating the isoperimetric problem for other polygonal prisms, not just rectangular, triangular, regular hexagonal, and circle ones. Furthermore, they can be extended to other three-dimensional figures such as cylinders, pyramids, cones, Platonic solids, and spheres. Additionally, the study and analyses presented in this thesis can be used as enrichment material for teaching geometry, particularly in the context of three-dimensional figures, in mathematics teaching.