Mathematics HL Internal Assessment written in 2018.

I got 20 out of 20 on both the predicted grade and final grade.

I investigated the physical properties of Toricelli’s Trumpet by using calculus (derivatives and integrals). Also, the paradox it has in real-life situations was delved in to make this paper more sound and logical.

**[Table of contents]**

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

**[Sample excerpts]**

**Exploring the Properties of Torricelli’s Trumpet**

**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. 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. Ultimately, I excavated the concept of Torricelli’s Trumpet in that its geometric properties lie precisely on the border between convergence and divergence. In addition, I have amassed an insight on the paradox between theoretical mathematics and the real world and that how comparably simple mathematical ideas can be put together to become an intriguing question for a mathematician.

**2. Introduction**

Torricelli first discovered Torricell’s Trumpet, or Gabriel’s Horn of infinite length with finite volume in the 17th century. Based on my research, such paradox causing a contradiction in our intuition is known as “Painter’s Paradox”. Relating back to previous explanation, the surface area of revolution of Torricelli’s Trumpet is infinite. This means that if we were to paint its surface, we must use an infinite amount of paint. Nevertheless, since the Gabriel’s Horn has finite volume, by pouring in a finite volume of paint, we should be able to fill its interior and thus paint the interior surface…

**Total number of pages**: 14 pages**Topic**: Exploring the Properties of Torricelli’s Trumpet**Subject**: Math**The file is in PDF format.**