Decathlon Mathematics Dr. Mary Garner, Professor Emeritus Kennesaw State University Instructor, Oglethorpe university

Decathlon Mathematics Algebra and Trigonometry Theme: World War II http://www.usad.org/Footer-Menu/Other/2015_2016Curriculum.aspx

WWII and Mathematics “The significance of the Second World War for the direction of – and growth in – scientific research in the USA was immense. In the case of mathematics the war gave rise to the emergence and establishment of new disciplines in applied mathematics as well as renewed interest and growth of research in related subjects in pure mathematics.” -- Tinne Hoff Kjeldsen , “New Mathematical Disciplines and Research in the Wake of World War II” from Mathematics and War Bridged the gap between pure mathematicians and engineers… Beginning of the military – industrial – university cooperation in research (Ga Tech)…

WWII and Mathematics Logistics Cryptography – The Enigma Code Weapon Use and Development Ballistics Torpedo launch control Manhattan Project Sonar and Radar

Logistics Operations Research -- the application of advanced analytical methods to help make better decisions. The terms management science and analytics are sometimes used as synonyms for operations research. “The job of OR groups during the war was not to invent new kind of weapons but to analyze what went on in the field and suggest ways to optimize the use of existing military equipment.” -- From Mathematics and War OR was “invented” in Britain in connection with responses to the growing German air force. Now so many universities have management science, operations research, …

WWII and Mathematics George Dantzig – father of linear programming. During WWII he worked on developing various plans or proposed schedules of training, logistics supply… planning problem formulated as a system of linear inequalities. http://ruby.fgcu.edu/courses/tharring/10183/m6_notes.htm

Logistics – Linear Programming SOLVE IT! “During World War II, the need for efficient transportation of supplies inspired mathematicians George Dantzig to develop linear programming. The LST was a ship used during WWII that carried 3-ton trucks and 25 ton tanks. The upper deck could carry 27 trucks, but no tanks. The tank deck could carry 500 tons , but no more than 33 trucks. Suppose an LST was to be loaded with as many tanks and trucks as possible, and at least three times as many trucks as tanks. What is the maximum number of tanks and trucks that could be loaded?” (From Algebra 2, by Larson, Boswell, Kanold, Stiff, published by McDougal Littell, page 169.)

Linear Programming

Logistics – Data Analysis Abraham Wald https://www.fastcodesign.com/1671172/how-a-story-from-world-war-ii-shapes-facebook-today

Cryptography Transposition cipher – rearranging the letters of the original message. Substitution cipher -- letters replaced by other symbols.

Cryptography SOLVE IT! Rearrange letters with a linear functions: f(x) = 3x – 10. Message: 8,35,35,50,-4,-7,26,26,-10,47,50,-7,44,50,47,-7,47,35,35,32

Cryptography f(x) = 3x – 10. Message: 8,35,35,50,-4,-7,26,26,-10,47,50,-7,44,50,47,-7,47,35,35,32, 71 Inverse of f(x) = 3x - 10 is f-1(x) = (x+10)/3. f-1(8)=6 f-1(26) = 12 f-1(35) = 15 f-1(-10) = 0 f-1(50) = 20 f-1(47) = 19 f-1(-4) = 2 f-1(44) = 18 f-1(-7) = 1 f-1(32) = 14 Message? Football starts soon!

Torpedo Launch Control

Torpedo Launch Control

Torpedo Launch Control

Torpedo Launch Control http://sites.temple.edu/topsecretrosies/

Torpedo Launch Control SOLVE IT! Suppose you are a torpedo officer aboard the USS Skipjack. Your submarine is conducting torpedo practice off the Florida coast. The target is 7200 meters from you on a bearing of 276 degrees (this means 276 degrees clockwise from north) and is steaming on a course of 68 degrees. You have long-range torpedoes that will go 6400 meters. Between what two bearings can you fire torpedoes that will reach the target’s path if you use the long-range torpedoes?

So angle CAE must be between 3.88 and 120.12 degrees.

Sonar and Radar Radar: Radio waves, transverse waves. Radar played a critical role in the British victory in the Battle of Britain in August of 1940. Hitler was planning an invasion of Great Britain. The British had only 800 aircraft to try to hold back over 3,000 German planes. The British victory was largely due to a series of radar stations that had been built along the south and east coasts of Britain in 1939. – From How Waves Helped Win the War: The uses of radar and sonar in WWII. Radar: Radio waves, transverse waves. Sonar: Sound waves, compression waves. (Sonar used by bats and dolphins.) From How Waves Helped Win the War: The uses of radar and sonar in WWII.

Sonar and Radar SOLVE IT! Problems 1. What is the range (distance to) an aircraft for which a radar pulse from a land radar station requires half of a millisecond (0.00050 seconds) to travel to the aircraft, be reflected and return? Remember, the velocity of radio waves is 300,000km/second. Use the formula: Round-trip distance = velocity x time 2. A destroyer on the ocean surface detects the reflection of a sonar “ping” (sound pulse) off a submarine a tenth of a second (0.10 seconds) after it was produced by the sonar device on the destroyer. What is the depth of the submarine? Remember, the velocity of sound in water is about 1.6km/second. Use the formula: Round-trip distance = velocity x time --From: From How Waves Helped Win the War: The uses of radar and sonar in WWII.

Resources How a Story From World War II Shapes Facebook Today by Mark Wilson, https://www.fastcodesign.com/1671172/how-a-story-from-world-war-ii-shapes-facebook-today New Mathematical Disciplines and Research in the Wake of World War II by Tinne Hoff Kjeldsen, from Mathematics and War published by Birkhauser Verlag in 2003. Access: http://link.springer.com/chapter/10.1007%2F978-3-0348-8093-0_6 Trigonometry, WWII Torpedoes, and a Museum Docent, posted 2011. http://mathscinotes.com/2011/03/trigonometry-wwii-torpedoes-and-a-museum-docent/ Submarine Torpedo Fire Control Manual http://www.maritime.org/doc/attack/index.htm Torpedo Data Computer http://www.maritime.org/doc/tdc/index.htm Torpedo Angle Solver http://archive.hnsa.org/doc/banjo/index.htm

More Resources How Waves Helped Win the War: Radar and Sonar in WWII -- A lesson plan written for the National WWII Museam by Louis Garcia. http://www.ww2sci-tech.org/lessons/lesson3.pdf Rediscovering WWII’s Female Computers http://www.cnn.com/2011/TECH/innovation/02/08/women.rosies.math/ Torpedo Data Computer http://maritime.org/tech/tdc.htm Harry Potter and the Coding of Secrets by Boon Liang Chua, published in Mathematics Teaching in the Middle School Volume 14, No. 2, September 2008. Wartime Trigonometry, by Albert Goetz, in Mathematics Teachers Vol. 109 Issue 8 April 2016.

Thank You! Please contact me at mgarner@kennesaw.edu!