The Average Area of a Triangle in a Parabolic Sector

1. The Average Area of a Triangle in a Parabolic Sector Allegheny Mountain Section Meeting of the MAA Indiana University of Pennsylvania April 6, 2013 Michael Woltermann Washington and Jefferson College

2. The Problem • Find the average area of the triangle formed by joining three points taken at random in (the surface of) a parabola whose base is b and altitude is h. • Problem 248 proposed by Enoch Beery Seitz in the Mathematical Visitor, 1880. • Solution published in 1893. • Senior MathTalk with Logan Elias (2012) at W&J, Fall 2012.

3. Published Solution • Solution appears to assume that the base is parallel to the directrix. • Avg = • Avg =

4. Avg = • Avg =

5. Is it True in this case? • b is not parallel to the directrix. • Is Avg = ? • Or

6. A property of parabolas • .

7. P=(v,u) • 0≤u≤h′ • -v′≤v≤v′ • Q=(x,w) • 0≤w≤u • -x′≤x≤x′ • R=(z,y) • w≤y≤u • -z′≤z≤z′

8. Area of Triangle PQR • P=(v,u) • Q=(x,w) • R=(z,y) • S=(t,y) • t= • Area(∆PQR) =

9. Average Area • Avg= • Factor out sin(ω), • And

10. The Average Area becomes • Avg = sin(ω)∙Seitz answer, or • Avg = • Or since sin(ω) = • Avg = • Link to: Excel Simulation

11. References • Problems and Solutions from The Mathematical Visitor 1877-1896, ed. By Stanley Rabinowitz, 1996, MathPro Press, Inc. • Archimedes, What Did He Do Besides Cry Eureka? By Sherman Stein, 1999, MAA. • http://archive.org/details/mathematicalvis00martgoog

12. Thank You!