Sample IA Work

Overview

Title Employing Mathematical modeling and optimization to minimize the use of plastic material in 1.5L Hanacka Water bottles

Research Question / Aim Minimizing the amount of plastic material needed to hold a fixed volume of water

Context / Personal Engagement Interest in sustainable practices and environmental issues Concerned by high levels of bottled water consumption and resulting waste Optimize bottle design to reduce plastic usage while maintaining functionality

Use of Mathematics Mathematical modeling of the bottle shape for accurate optimization Application of integration to calculate the volume and surface area of solids Constrained optimization using the Lagrange multiplier method

Constrained Optimization

The usage of complex numbers in analyzing Alternating Current (AC) Resistor-Inductor-Capacitor (RLC) circuits to find impedance and phase angle

Session: May 2023

Page count: 20

Introduction and Aim

The exploration aim is to explore mathematical methodologies used to analyze AC circuits, and to demonstrate that mathematical methods using complex numbers may be more efficient than those using only real numbers, even for real-life applications of mathematics. Findings shall be verified experimentally to determine whether the theoretical methods used are accurate. The focus is on impedance and phase angle for resistors, inductors, and capacitors connected to an AC power supply. This analysis explores a multitude of topics from the IB AA HL curriculum, including Euler’s form of complex numbers, Maclaurin series, Argand diagrams, vectors and phasors, trigonometric functions because of AC’s periodic motion, differentiation, integration, as well as differential equations to highlight the need for complex numbers in AC circuit analysis. This investigation will first introduce the mathematical physics required, then why complex numbers should be used, the impedances of the three individual circuit components, their combined impedance represented visually, a worked example problem using our findings, and finally, an experimental verification.

I chose this exploration because I have a passion for Physics, especially for the electromagnetism topic. I have a keen interest in circuit building and have always wanted to know more about the electronic devices we use in our daily lives. Personally, I find it fascinating how imaginary numbers may make real life applications of mathematics easier, especially in circuits which is arguably more focused on the experimental as opposed to the theoretical part of physics. To see whether this seemingly contradictory idea of using imaginary numbers in real-life applications is valid, I will build my own circuit to compare the experimental value to my manually-calculated value.

X_Complex Number

Mathematically Modelling Household Metal Containers: An Investigation into the Most Suitable Container(s) to Store a Hot Beverage

Session: May 2022

Page count: 16

Introduction and Rationale

My family comprises coffee and tea drinkers - myself included - and we enjoy drinking our beverages warm, especially in the cooler months of the year. Often times, we find that the beverage cools quickly after preparation, allowing us to only consume it at a suboptimal, lukewarm temperature. I wanted to figure out a method to both combat the quick cooling of the drinks and ensure that the beverage was as warm as possible during consumption, and I thus decided to mathematically model different metal containers in my house in order to arrive at a conclusion for the most suitable container to keep a drink warm (i.e. one that does not lose heat quickly), by calculating and collating their respective surface area to volume ratios. The surface area to volume ratio is crucial to a multitude of fields - notably Biology – generally in terms of the impact a particular body’s ratio has on its temperature and transfer of heat. For instance, elephants have developed big ears and wrinkled skin in order to increase their surface area to volume ratio1, allowing for greater dissipation of heat during warmer weathers. This is because the larger the surface area to volume ratio, the greater the allowance for heat dissipation, since this allows for greater area for heat energy to escape2. I thus decided to apply the same concept to the metal containers in my exploration. For the purpose of the investigation, the three containers that I chose to explore are the three most common metal containers/cups that are used in my house, which allows me to generalise my findings or conclusion to all other containers of the exact same model in the house. It is important to note the investigation does not condemn any manufacturers, and all chosen containers have been displayed for the purpose of academic and mathematical discourse and exploration.

Aim and Methodology

X_Container Storage

X_Distribution - Curve & Coefficient

X_Modelling - Differential Equation

Modeling and Optimization into different factors of the safety nets to minimize the casualties from a falling brick from a construction site.

1. Introduction

These days, skyscrapers are mostly built in cities for various purposes. However, there are possible dangers of freefalling bricks or metal materials that cause serious damage to workers and even passers-by. A normal force that does the slightest damage in the human head is 84 grams, but considering the weight a normal construction brick,2.04 kg(CivilSir, 2023), the brick itself is a deadly weapon for the passengers below. Therefore, to prevent casualties, safety nets are implemented in building walls in countries such as Singapore, England and Japan. Yet in Vietnam safety nets are not mandatory, domestic construction firms purposely exclude the safety nets to save money. This fails to assure the safety of the peasants and workers themselves.

After secondary research, I discovered how vulnerable the construction fields in Vietnam are, since it could not assure the safety of the workers and the passengers compared to other nations that implement nets, even sold in the internet widely, while this is not a common item that is found explicitly in Vietnam. Therefore, my IA focuses on dealing with the issue of safety. I will think of possible dimensions, angles, thickness and material for safety nets so that this could prevent hazardous issues in the future. In order to understand where the brick fell, I found a construction site with an unfinished building, which could increase the credibility of my result rather than scaling down the experiment. After the experiment, my plan is to find the dimension that has high efficiency by plotting a reference point on the position where the brick fell and draw a normal distribution model to find out the range of the distance where it has the highest percentage.

X_IA Work