IB Math HL IA_Curve Length & Curvature (7점)

How can I use the concept of curve length and curvature to find the contact lens fit on myself?

Introduction

Since Grade seven, I started to wear contact lenses. But some lenses did not fit on my eye and made me feel irritated. And in ninth grade, I was able to find a contact lens that fits my eye. Although it is the same lens, some people feel it comfortable to wear, but others don’t. And this made me interested in finding the optimum lens that fit each person—and finding optimal contract lenses for myself by investigating the curve length of my eye and contact lenses available in stores. This is practicable because every contact lens has a different curve length.

My personal experience helped me construct the research question: How can I use the concept of curve length and curvature to find the contact lens fit on myself?

I decided to use the linear distance formula that I learned in IB math class to find the curve length. I thought that curve length has the same concept as the linear distance formula because it is a formula that solves the distance between two points. But If I only use this formula, it would give me the linear length of two points. So I decided to expand this formula using mathematics that I learned in IB math.

However, after investigating the curve length, I realized that the curve length and the curvature affect the wearability. Even though every lens has the same diameter, they had different curve lengths. Some were shorter than others, like in the diagram below…