Archimedes’s Method

MATH 242 practice problems

Learning Goals: Gain proficiency with programming in Mathematica. Develop and implement an algorithm to solve a mathematical problem, producing understandable results.

Your task: Complete the following exercises by 5pm on Wednesday, February 11. Try to complete this before class on Wednesday, but you can ask last-minute questions on Wednesday before turning it in. You can do the work in this notebook, below each numbered item.

Grading: Upload your solutions as a Mathematica notebook to the Archimedes's Method assignment on Moodle. If you complete each problem, then you will earn a Satisfactory grade.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, or email Prof. Wright.

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What assistance did you receive and what resources did you use (if any) while working on this assignment?

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Problem 1

In class we observed that the number of correct digits of π produced by Archimedes’s method appears to increase linearly with the number of iterations. Find an equation of a line that describes this linear relationship. You don’t need any fancy methods for this: just choose two data points and find the equation of the line that connects them. Does it matter which two points you choose? Explain!

Problem 2

In class, we approximated π using polygons with 4, 8, 16, 32, ... sides. Now modify the code to produce approximations of π starting with an inscribed hexagon, and then repeatedly doubling the number of sides. That is, you will use inscribed polygons with 6, 12, 24, 48, ... sides. Produce both a table and a plot showing the accuracy of your approximations (like we did in class).