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

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.

PUBLISHED SOLUTION

Solution appears to assume that the base is parallel to the directrix.

Avg =

Avg =PArea

140

11

Avg =

Avg =

dV

dVarea(...)

3

32

)(6

bh

dVPQRarea

IS IT TRUE IN THIS CASE?

b is not parallel to the directrix.

Is Avg = ? Or

?140

11PArea

A PROPERTY OF PARABOLAS

.

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′

h

ybz

h

wbx

h

ubv

2,

2,

2

AREA OF TRIANGLE PQR

P=(v,u) Q=(x,w) R=(z,y) S=(t,y)

t=

Area(∆PQR) =

wu

yuxvv

))((

)sin(2

1 wuzt

AVERAGE AREA

Avg =

Factor out sin(ω),

And

h v

v

h x

x

h z

z

h v

v

u x

x

u

w

z

z

dudzdydxdwdv

dudzdydxdwdvwuzt

0 0 0

0 0

)sin(21

6

THE AVERAGE AREA BECOMES

Avg = sin(ω)∙Seitz answer, or

Avg =

Or since sin(ω) =

Avg =

Link to: Excel Simulation

),sin('210

11 bh

,h

h

PAreabh140

11

210

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

THANK YOU!