Decathlon Mathematics
DR. MARY GARNER, PROFESSOR EMERITUS
MGARNER@KENNESAW.EDU
Decathlon Mathematics
General mathematics, geometry and differential calculus
Theme: New Alternatives in Energy: Ingenuity and Innovation
Book: Anne Dillard’s Pilgrim at Tinker Creek
Decathlon Mathematics
“I
feel as though I stand at the foot of an infinitely high staircase,
down which some exuberant spirit is flinging tennis ball after tennis
ball, eternally, and the one thing I want in the world is a tennis ball.”
― Annie Dillard, Pilgrim at Tinker Creek
“Push it. Examine all things intensely and relentlessly.”
― Annie Dillard, Pilgrim at Tinker Creek
Decathlon Mathematics
“She brings into her discussions knowledge from the whole spectrum
of thought - referencing physics, literature, numerous religious
traditions, anthropology, medicine and folklore -- just to name a few
of her sources.”
--http://sandra.stahlman.com/dillard.html
This Presentation
One Geometry problem
One Calculus problem
with an historical view,
always considering infinity, and
applied to solar energy
Geometry/Calculus
Archimedes – 3rd century B.C.
https://www.cs.drexel.edu/~crorres/Archimedes/contents.html
Archimedes proved, among many other geometrical results, that the
volume of a sphere is two-thirds the volume of a circumscribed
cylinder.
Method of Exhaustion
An equilateral triangle is
inscribed in a circle with radius r.
What is the area of each of the
isosceles triangles shown?
What is the area of the
equilateral triangle?
Method of Exhaustion
3 2
𝑟 sin 𝜃
2
Method of Exhaustion
An square is inscribed in a circle
with radius r.
What is the area of each of the
triangles shown?
What is the area of the square?
Method of Exhaustion
A regular pentagon is inscribed in a
circle with radius r.
What is the area of each of the
triangles shown?
What is the area of the pentagon?
Method of Exhaustion
A regular hexagon is inscribed in a
circle with radius of r.
What is the area of each of the
triangles shown?
What is the area of the hexagon?
Method of Exhaustion
What’s the pattern?
yB = 2.08
yC = 2.51
yD = 2.83
yE = 2.97
yF = 3.02
yG = 3.06
yH = 3.11
12
10
f(x) =
() (
x
∙sin
2
2∙3.14159 radians
x
)
8
6
f(500) = 3.14
4
2
B
3π
2π
π
π
2
4
6
C
2π
3π
4π
H
G
F
E
D
5π
6π
7π
8π
9π
More Archimedes
Infinite Secrets … NOVA
Geometry/Calculus and Solar Energy
http://www.mathaware.org/mam/2013/
sustainability/Solar-Panel-Lesson.pdf
http://mathaware.org/mam/2013/sustain
ability/Solar-Panels-Energy-and-AreaUnder-Curve-Teacher-Guide.pdf
Geometry/Calculus and Solar Energy
Energy is usually measured in kWh which stands for kilowatt hours.
Energy = Power x Time
Power is the rate at which energy is flowing through the system( 1 watt = 1 joule per sec). So the
power associated with ten 100-watt bulbs all on at the same time is 10 x 100 = 1000 watts or 1
kW.
The energy used by ten 100-watt bulbs, burned for 3 hours would be 3 kWh.
Power is the derivative of energy and energy is the integral of power.
Geometry/Calculus and Solar Energy
https://enlighten.enphaseenergy.com/pv/public_systems/faZD132956
Geometry/Calculus and Solar Energy
How fast is power
Increasing at t = 9 hours?
Geometry/Calculus and Solar Energy
Geometry/Calculus and Solar Energy
2.1 − 1.75
= .07
14 − 9
Geometry/Calculus and Solar Energy
2.5 − 1.75
= .19
13 − 9
Geometry/Calculus and Solar Energy
2.7 − 1.75
= .32
12 − 9
Geometry/Calculus and Solar Energy
2.1 − 1.75
= .35
10 − 9
Geometry/Calculus and Solar Energy
What about
Energy?
Energy = Power x Time
Geometry/Calculus and Solar Energy
What about
Energy?
Energy = Power x Time
= area under the curve
Geometry/Calculus and Solar Energy
What about
Energy?
Energy = Power x Time
= area under the curve
Geometry/Calculus and Solar Energy
What about
Energy?
Energy = Power x Time
= area under the curve
Geometry/Calculus and Solar Energy
https://enlighten.enphaseenergy.com/pv/public_systems/faZD132956
Geometry/Calculus and Solar Energy
http://www.rawlemon.com/
Decathlon Mathematics
“How is it possible that mathematics, a product of human thought that is
independent of experience, fits so excellently the objects of reality?
—Albert Einstein
The "great book" of the universe is written in the language of mathematics, he
(Galileo) famously declared, and unless we understand the triangles, circles, and
other geometrical figures that form its characters, he wrote, "it is humanly
impossible to comprehend a single word of it [and] one wanders in vain through
a dark labyrinth.“
-- http://www.pbs.org/wgbh/nova/physics/describing-nature-math.html
"Just as music is auditory patterns that the human mind finds pleasant,
mathematics captures patterns that the universe finds pleasant, if you like—
patterns that are implicit in the way the universe works.“ – Keith Devlin
Thank you!
“Push it. Examine all things intensely and
relentlessly.”
― Annie Dillard, Pilgrim at Tinker Creek
And what will you find?
Mary Garner
mgarner@kennesaw.edu
Thank you!
“Push it. Examine all things intensely and
relentlessly.”
― Annie Dillard, Pilgrim at Tinker Creek
And what will you find?
Mathematics!
Mary Garner
mgarner@kennesaw.edu