Perluasan Masalah Isoperimetrik pada Bangun Ruang

Andri Setiawan, Denny Ivanal Hakim, Oki Neswan

Abstract


In this paper, several extensions of the isoperimetric problem in solid figures are explored, focusing on oblique and right prisms with rectangular, right-angled triangular, and regular hexagonal bases. The objective of this research is to find the prism with the largest volume while keeping the surface area constant. Through manipulations of algebra and simple trigonometry, evidence is obtained that a right prism provides a larger volume than an oblique prism if their surface areas are equal. By utilizing partial derivatives of a two-variable function and the Lagrange multiplier method, conditions for the side lengths are derived to obtain the prism with the maximum volume. The results show that a cube is the solution to the isoperimetric problem, meaning it has the largest volume among prisms with rectangular bases, while for the isoperimetric solution on prisms with right-angled triangular bases, the base of the prism must be an isosceles right-angled triangle. A regular hexagonal prism has a larger volume than prisms with rectangular and right-angled triangular bases if their surface areas are the same.

Keywords


Isoperimetric; Extension of Isoperimetric Problem; Solid Figure; Right Prism; Oblique Prism

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References


N. D. Kazarinoff, Geometric Inequalities. New York: The Mathematical Association of America, 1961, doi: 10.5948/UPO9780883859223.

R. F. Demar, “A simple approach to isoperimetric problems in the plane,” Mathematics Magazine, vol. 48, no. 1, pp. 1–12, 1975, doi: 10.1080/0025570X.1975.11976430.

V. Blasj ˚ o, “The isoperimetric problem,” The American Mathematical Monthly, vol. 112, no. 6, pp. 526–566, 2005, doi: 10.1080/00029890.2005.11920227.

C. Rollin, A Short History of Carthage. Oregon: Perennial Press, 2018, [online] available:https://books.google.co.id/books?id=oLJ4DwAAQBAJ.

R. Fitzgerald, The Aeneid Virgil. New York: Vintage Books, 1983 [online] available: https://archive.org/details/aeneid00virg.

T. L. Heath, A history of Greek mathematics. Oxford: Clarendon Press, 1921, [online] available: https://archive.org/details/historyofgreekm02heat/mode/1up.

I. Bulmer-Thomas, Selections illustrating the history of Greek mathematics. Cambridge: Harvard University Press, 1939, [online] available: https://archive.org/details/selectionsillust02bulmuoft.

G. Polya and G. Szego,˝Isoperimetric Inequalities in Mathematical Physics. (AM-27). Princeton: Princeton University Press, 1951, [online] available: http://www.jstor.org/stable/j.ctt1b9rzzn.

I. Chavel, Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives. new York: Cambridge University Press, 2001.

U. Stammbach, “A letter of hermann Amandus schwarz on isoperimetric problems,” The Mathematical Intelligencer, vol. 34, no. 1, pp. 44–51, 2012, doi: 10.1007/s00283-011-9267-7.

D. C. Alexander and G. M. Koeberlein, Elementary Geometry for College Students, 7th ed. Boston: Cengage Learning, 2019.

W. F. Kern and J. R. Bland, Solid mensuration with proofs, 2nd ed. New York: John Wiley and Sons, Inc, 1938.

A. C. F. L. I. E. Leonard, J. E. Lewis and G. W. Tokarsky, Classical geometry : Euclidean, transformational, inversive, and projective. New York: John Wiley and Sons, Inc, 2014.

S. Karmakar and S. Karmakar, Multivariate Calculus. London: CRC Press, 2023, doi: 10.1201/9781003407874.

E. P. D. Varberg and S. Rigdon, Calculus, 9th ed. New Jersey: Prentice Hall, 2006.

A. G. S. Ventre, Calculus and Linear Algebra. Cham: Springer International Publishing, 2023, doi: 10.1007/978-3-031-20549-1.




DOI: https://doi.org/10.34312/jjom.v5i2.20534

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