Math HL IA

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2018에 작성했던 Mathematics HL Internal Assessment입니다.

Predicted grade와 실제 Final grade 모두 20점 만점을 받은 검증된 IA입니다.
Toricelli’s Trumpet의 물리적 특성을 미적분을 사용하여 탐구해보았고, 그것이 가지고 있는 파라독스, 그리고 현실세계에 적용될 때 있어서의 괴리까지 함께 풀어보았습니다. Calculus가 이 논문에 쓰인 가장 중요한 컨셉이며, 학생의 수준에서 이해하기 어려울 정도로 난해한 식을 쓰지 않고서도 충분히 수준높은 식을 도출해 냄으로써 난이도적 측면에서도 높은 평가를 받을 수 있었습니다.

수학에 대해 두려움이 있거나 막연함을 품고 있는 학생들과 또는 어떻게 해야 효율적으로 높은 점수를 얻을 수 있는지 고민하는 모든 학생들에게 충분히 도움이 될 수 있을 문서이니 믿고 구매해주시길 바랍니다.



Table of contents
1. Rationale —————————————————————————————– 2
2. Introduction ————————————————————————————– 2
3. How to build Torricelli’s Trumpet ———————————————————— 3
4. Volume of Torricelli’s Trumpet ————————————————————— 4
5. Surface area of Torricelli’s Trumpet ———————————————————- 5
6. What is Gabriel’s Wedding Cake ————————————————————- 6
7. Working out the volume of Gabriel’s Wedding Cake ————————————– 6
8. Finding the surface area of the Gabriel’s Wedding Cake ———————————- 8
9. Investigation of the Painter’s Paradox ——————————————————- 9
10. Conclusion and Evaluation ——————————————————————- 12
11. Bibliography ———————————————————————————– 14



1. Rationale
For this investigation, I had a chance to explore a Torricelli’s Trumpet. By definition,
Torricelli’s Trumpet is a geometric figure created by the solid of revolution of y = 1/x graph
about the x-axis at intervals of one to infinity[1]. As a trumpet player, I often put my arm inside
the trumpet so that I can clean its interior. It was difficult to clean inside, as the space became
so small. In that moment, I became curious if the hole of the trumpet becomes even smaller
than the air particle, would there be any sound coming out. Later, I searched the internet about
such trumpet and surprisingly, such trumpet actually existed; with a term called Torricelli’s
Trumpet. I found out that Torricelli’s Trumpet has infinite surface area yet finite volume. This
finding was perceived to be counterintuitive as I thought finite volume and the finite surface
area must be correlated. Accordingly, this exploration is primarily focused on mathematically
justifying the properties of Torricelli’s Trumpet and debugging such counterintuitive notion;
while also examining a much simplified real-life example, a Gabriel’s Cake. By its very nature,
I have applied mathematics in the course – mainly integration and convergence tests…


  • 총 페이지수: 14 pages
  • 과목: Mathematics
  • 주제: Exploring the Properties of Torricelli’s Trumpet
  • The file is in PDF format 
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Math HL IA

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