Activity Overview
Students use calculus to determine the maximum size of the iris (circle) that just fits inside an outline of the eye defined by two bell shaped curves. The activity uses some basic differential calculus, introduces simple substitutions to eliminate variables and handy techniques for simplifying problems. The problem is much easier than it looks!
Objectives
Use calculus to determine the maximum area of a circle that just fits inside another curve. The combined curves appear as an eye adding some visual interest to the problem. The activity also introduces some techniques that students may use to help solve other, more complicated problems.
Vocabulary
- Differential Calculus
- Exponential function
- Gradient of a function
- Substitution (algebra)
- Simultaneous equations
About the Lesson
The calculus in this activity is relatively straight forward, however students need to be careful. There are several approaches to solving this problem and almost as many ways to overlook simple assumptions. It is a necessary condition that the gradient of the two curves (iris and eye boundary) are the same where the maximum iris area occurs, but these are not sufficient conditions. An accurate and animated visual help solve the problem as well as introducing some straight forward techniques. With or without CAS, the algebra and calculus in this problem is readily accessible.