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!