The computation of π is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research. -Len Berggren, Jonathon Borwein, and Peter Borwein, Pi: A Source Book
π can be defined as the ratio of the circumference to the diameter.
π is prevalent in numerous mathematic problems like lengths of arcs, areas of circles and ellipses, volumes of different solids, and appears in numerous formulas (Pi | mathematics).
The formula for the length of arcs is Arc Length =θ π/180 r.
π also is in the formula for area of a circle A=πr^2, area of an ellipse A= major axis length*minor axis length*π, and Circumference of a circle C= πd (Breyer).
The most common use of π is when dealing with circles. π relates to circumference, diameter, and area of circle.
Most people have been exposed to the formulas A= πr^2 and C= πd at some point in their lives.
A lot of people, ancient and current have used these relationships to calculate π (Bogart).
As stated before, π is the ratio of the circumference to the diameter, but the significance of that is that it is true for all circles.
Physicists have found π a lot in nature. They have found it in the disks of the moon, in a rainbow, in the pupil of an eye, in the double helix structure of DNA, in ripples in water, and even in colors and music(Witcombe).
Rivers meander and the meanders are described by the length of its path divided by the distance from the source of the river to the ocean.
This is called its sinuosity and the average river has the sinuosity of 3.14 (Breyer).
The pupil of a human eye is nearly a perfect circle and is designed to work in bright light.
The arc of a rainbow is the second closest thing to a perfect circle in nature, losing only to ripples in water.
The arc of a rainbow will never been seen as more than a semicircle to humans unless it is viewed from the height of a raincloud with the sun behind the human.
Ripples in water are the most perfect circle that can be found in nature.
The interesting thing is it doesn't matter what shape is thrown into the water, the ripples always become circular (Poon).
A lot of people have calculated the digits of π into the millions. They use super computers to do this, and people continue to do this.
The calculation of π is used as a stress test for computers (50 Interesting Facts About Pi).
The world record for reciting π belongs to Lu Chao of China, who recited π to more than 67,000 decimal places.
Even if she said the digits quickly that would be at least 6 hours of reciting (Hom). People have created poems and sayings to help them remember the digits of π.
The length of the word is representative of each consecutive digit in π. An example of a poem is May I have a large container of butter today (Pi (π)).
Mike Keith has even written a book using the same constraint that π poems do.
The book was called Not a Wake and was the first book to ever be written in Pi-lish (Hom).
One of my favorite applications of π came from my curiosity as a child.
As a kid I loved ice cream, it was probably my favorite food, and I especially loved it in a cone.
I wanted to eat it for breakfast, but was seldom able to. I loved it so much that my parents started to limit my ice cream consumption.
This limitation got me thinking about saving my ice cream cone for later.
My problem was that if I put a scoop of ice cream onto a cone, when the ice cream melted, would I be able to save it for later?
Or would the cone overflow and would I have created a mess?
This question puzzled me for a long time because sometimes it would work, but other times it would overflow.
I didn't know how to determine if it would overflow. Eventually I learned about π and volumes of cones and spheres, this was a game changer.
I started playing around with different sized scoops and different sized cones and that has inspired the following problems.
1. Your friend gave you an ice cream cone, but you aren't hungry right now.
You decide that you want to save it for later, but your friend tells you that you can't because it will melt and the cone will overflow and it will spill everywhere.
Is that true? You take out your ruler and measure the height of the cone is 6 inches and the radius of the cone is 2 inches.
The radius of the ice cream scoop is 3 inches. Can you save your ice cream for later or will your cone overflow?
2. You bought an ice cream cone for 3 dollars.
The height of your cone is 4 inches, and the radius of your cone is 2 inches.
The radius of your ice cream scoop is 3 inches. Your friend also bought an ice cream cone for 3 dollars.
The height of her cone is 6 inches and the radius of the cone is 3 inches. The radius of her ice cream scoop is 2 inches.
Both cones are filled with ice cream plus the scoop on top. Who got the better deal and got more ice cream?
Solution to problem 1.
The formula for volume of a cone is V= 1/3 [πr^2]h and the formula for volume of a sphere is V=4/3 πr^3.
If the cone's height is 6 inches and the radius is 2 inches then the volume of the cone is V=1/3 [π 2^2 ]6 = 24/3 π = 8πin^3≈25in^3.
Now we need to determine the size of the ice cream scoop. The volume of that is V=4/3 π2^3 = 32/3 π in^3≈34in^3.
So 34in^3 is bigger than 25in^3, so if it melted it would overflow. I guess you'd better eat your ice cream now.
Solution to problem 2.
Using the formulas for volume of a cone and volume of a sphere listed above in the solution to problem 1.
So the total volume of ice cream in your cone would be V= 1/3 [πr^2 ]h + 4/3 πr^3 = 1/3 [π2^2 ]4+ 4/3 π3^3= 16/3 π+ 108/3 π= 124/3 π ≈ 130in^3.
The total volume of ice cream in your friend's cone would be V= 1/3 [πr^2 ]h+ 4/3 πr^3 = 1/3 [π3^2 ]6+ 4/3 π2^3= 54/3 π+ 32/3 π= 86/3 π ≈ 90in^3.
Since you both got them for 3 dollars, I would say you got a way better deal!
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