2018 Excellence in Mathematics Contest Team Project

Transcription

1 2018 Excellence in Mathematics Contest Team Project School Name: Group Members: 2018 Scott Adamson

2 Reference Sheet Formulas and Facts You may need to use some of the following formulas and facts in working through this project. You may not need to use every formula or each fact. A bh C 2l 2w Area of a rectangle Perimeter of a rectangle Area of a circle 1 y y C 2 r A bh m 2 x x Circumference of a circle Area of a triangle Slope A r inches = 1 foot 5280 feet = 1 mile 3 feet = 1 yard 1 mile = 1.61 kilometers 2.54 centimeters 1 inch 1 year = 365 days 1 kilogram 2.2 pounds 1 ton = 2000 pounds 1 gigabyte = 1000 megabytes 1 mile = 1609 meters 1 parsec = 19,000,000,000,000 miles 1 gallon 3.8 liters 1 sq. yd. = 9 sq. ft 1 cu. ft. of water 7.48 gallons 1 ml = 1 cu. cm V r h V lwh V r 3 Volume of cylinder Volume of rectangular prism Volume of a sphere Lateral SA = 2 r h Lateral surface area of cylinder b b 2 4ac x 2a Quadratic Formula tan sin cos 1 parsec = 3.25 light years 1 square mile = 640 acres tan sin cos 2

3 TEAM PROJECT 2018 Excellence in Mathematics Contest The Team Project is a group activity in which the students are presented an open ended, problem situation relating to a specific theme. The team members are to solve the problems and write a narrative about the theme which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content, clarity of explanations, and creativity in their narrative. Part 1 Introduction By Andrea Dresdale Stephen Iervolino May 25, 2017, 4:16 AM ET 'Star Wars' opened on this day in 1977 On May 25, 1977, fans were introduced to a galaxy far, far away. While 20th Century Fox had little faith a movie called "Star Wars" would be a hit, the space opera from young filmmaker named George Lucas became a worldwide cultural phenomenon. Forty years, and a handful of sequels, prequels, and follow-ups later, the multibillion-dollar franchise is a popular as ever. After a successful re-launch with J.J. Abrams' "Star Wars: The Force Awakens" in 2015, fans also flocked to the first standalone "anthology" movie, 2016's "Star Wars: Rogue One." In December, we'll see the eighth chapter of the core franchise, "The Last Jedi." Next year comes a film about the early days of Han Solo, scheduled for a May 25, 2018, release, followed by "Episode IX" on May 24, There's also at least one other rumored standalone film about the early adventures of another fan favorite character. Mark Hamill, who plays Luke Skywalker in the franchise, thanked fans for their years of dedication at a panel at the Star Wars Celebration in Orlando last month. "It's almost moving when you really think about how we've affected so many people, inspired so many people," he told fans. "[I]t's generational -- that it's handed down: the original fans are grown, with children of their own." He added, "It's a gift, and something I will never take for granted." Of course, any "Star Wars" anniversary reminds fans that Carrie Fisher, who played Princess Leia -- later General Leia -- is no longer with us. In the latest issue of Vanity Fair -- which features different "Star Wars" stars on four different covers -- Lucasfilm president Kathleen Kennedy says the late actress was determined to be the hero of "Episode IX," coming in "The minute she finished [shooting 'Episode VIII'], she grabbed me and said, I d better be at the forefront of 'IX!' Because Harrison [Ford] was front and center on 'VII,' and Mark [Hamill] is front and center on 'VIII,'" says Kennedy. "She thought 'IX' would be her movie. And it would have been. As for Hamill, he tells Vanity Fair, "I can't say that phrase... Carrie's name and then the 'd-word.' Because I think of her in the present tense. Maybe it s a form of denial, but she s so vibrant in my mind, and so vital a part of the family, that I can t imagine it without her. Hamill also reveals that he and Fisher were "really attracted to each other" back in the day, and even had "makeout sessions," until they decided to pull back for fear of harming their relationship. In the "Star Wars" universe, of course, Luke and Leia were twins, but in that original film, they didn't know that. 3

4 Part 2 Best Shot with Blaster Most Star Wars fans acknowledge that Stormtroopers are not very accurate when shooting their blasters. One Star Wars fan has claimed to have re-watched all blaster battles in Star Wars and collected the data shown below. This fan claims to have only counted shots where the result of a hit or a miss was clear. Any shot that we don t see land one way or the other is simply not counted. ( Based on the data provided ( make a convincing mathematical argument for which Star Wars character is the most accurate shot with a blaster. You must choose one character and present a clear mathematical argument for why you claim this character is the best at shooting a blaster. 4

5 Part 2 continued The percentages are provided below. Note that students may argue that Finn is the best blaster shooter but other students may look at the number of shots taken to make a different argument. The goal is to consider the strength of the mathematical argument and the clear communication of this argument. 5

6 Part 3 Star Wars Characters Hidden Links Between Star Wars Characters Revealed With Mathematics Over the course of the seven Star Wars films, an intergalactic tale full of vivid, mythical characters has played out, each with their own complex backstories. This rich detail means that even die-hard fans can have trouble making sense of it all. With this in mind, a group of researchers at Ecole Polytechnique Federale De Lausanne (EPFL) in Switzerland, led by Ph.D. student Kirell Benzi, decided to attempt to categorize this forest of information by developing a series of novel algorithms. The algorithms, known as web scrapers, were designed to pick up on unique characters within the contents of Wookipedia, a Star Wars version of Wikipedia. Information about when they lived during the history of the saga, and any major events that happened throughout their lives, was absorbed. By putting this information into graph form, the researchers uncovered a wealth of new data that was previously hidden. Significantly, web scrapers carefully looked for all the times when two or more characters crossed paths, either directly or indirectly. Considering the saga spans 36,000 years of time, this was no easy task. Remarkably, there are 21,647 characters, with 19,612 of them having names and some degree of backstory at the very least. Out of these characters, it appears, perhaps unsurprisingly, that 78 percent of them are human. Twi Leks, Rodians and Wookiees make up the 1st, 2nd and 3rd most populous galactic minorities. All of these lifeforms are spread over 294 planets, over which 1,367 Jedi and 724 Sith have at some point lived. For much of the younger part of the timeline, there were only two Sith simultaneously alive a master and an apprentice so it s not surprising that, overall, they re outnumbered. Some of the characters do not have a designated timeline, but as the algorithms are able to work out who they most likely interacted with, they can be placed at their probable points within the saga s history. Out of all these characters, 7,500 play an important role, in that they are highly connected to the lives of others. Accompanying the above article is the following graph. Your task is to a) determine the total number of each of the categories of Star Wars characters b) determine the percentage associated with each of the categories of Star Wars characters c) determine the measure of the central angle associated with each section of the chart shown. Please record all of your results in the table on the next page. 6

7 Part 3 continued Star Wars Species Total Number (round to the nearest whole number) Percent of Total (round to the nearest tenth of a percent) Measure of Central Angle in Pie Chart (round to nearest degree) 16906/ o Human 1212/ o Twi Lek 909/ o Rodian 714/ o Wookie 411/ o Bothan 368/ o Yuuzhan Vong 368/ o Trandoshan 303/ o Zabrak 303/ o Sullustan 238/ o Duros Note: Article uses two total numbers for the number of Star Wars characters: 21,647 and 19,612. Also, due to rounding, the total percentage is 100.4% and the total number of characters is 21,732 or 19,691. 7

8 Part 4 Comparing Heights Harvard Bridge spans the Charles River linking Boston and Cambridge. In 1958 Lambda Chi Alpha took 5' 7" MIT freshman pledge Oliver R. Smoot, Jr. and flipped him head over heels the entire length of the bridge. Every ten smoots they calibrated the bridge, painting marks. The bridge was found to be exactly smoots plus an ear. Successive pledge classes repainted the markings. Convert the measurement of the Harvard Bridge to chewbaccas, ewoks, and porgs. You may show all your work on the next page as needed. Round all answers to the nearest tenth of a unit. Chewbacca is 7 3 tall. An Ewok is about 1 meter tall. A Porg is 10 inches tall Not to scale The Harvard Bridge is smoots long. Harvard Bridge is chewbaccas long. The Harvard Bridge is ewoks long. The Harvard Bridge is porgs long. 8

9 Part 4 continued 9

10 Part 5 Star Wars Toys According to the Netflix series, The Toys that Made Us, Star Wars movies also created an incredibly huge toy industry. In the Netflix series, it is claimed that if the plastic from all of these toys was melted down, a giant toy Wookie could be created that would be taller than the Empire State Building in New York City. They also claim that in the 1970s and 1980s, 22 million Star Wars toys were sold each year. 1. According to the Empire State Building has a volume of 37 million cubic feet. Suppose 1,000,000,000 Star Wars toys have been sold. What is the average volume of each toy? Express your answer in cubic inches and cubic centimeters (milliliters) cubic feet cubic inches cubic centimeters/milliliters 2. Use the rate of sales reported in The Toys that Made Us to determine the length of time needed to sell 1,000,000,000 toys. Communicate your solution using a graph and formula in addition to any calculations needed. It takes about 45.5 years to sell 1,000,000,000 toys at a rate of 22,000,000 toys per year. If t is the number of years since 1977 and F is the number of toys sold in millions, then the linear model F = 22t can be used to model the situation. 10

11 Part 6 The Amazing Millennium Falcon In Episode IV, A New Hope, there is a well-known scene with Han Solo and Obi-Wan Kenobi where Han says, You've never heard of the Millennium Falcon? It's the ship that made the Kessel Run in less than twelve parsecs. According to The Kessel Run was one of the most heavily used smuggling routes in the Galactic Empire. Han Solo claimed that his Millennium Falcon "made the Kessel Run in less than twelve parsecs". A parsec is a unit of distance, not time. Solo was not referring directly to his ship's speed when he made this claim. Instead, he was referring to the shorter route he was able to travel by skirting the nearby Mawblack hole cluster, thus making the run in under the standard distance. By moving closer to the black holes, Solo managed to cut the distance down to about 11.5 parsecs. The smuggler BoShek actually beat Solo's record in his ship, Infinity, but without cargo to weigh him down. A few months later, Han Solo beat both his own and BoShek's records in a run he made with Luke Skywalker. To get a sense of the size of distances in space, answer the following hypothetical questions. 1. How long would it take for a car to make the 12 parsec Kessel Run if it traveled at a constant speed of 85 MPH? trillion miles 24 hours per day 365 days per year 85 MPH Using 1 parsec = 19 trillion miles, it would take over 306,204,673.7 years. 2. At what average speed, in MPH, would a vehicle need to travel in order to make the 12 parsec Kessel Run in 7 days? trillion miles 7 24 hours The vehicle would need to travel at an average speed of 1,357,142,857,000 MPH. 11

12 Part 6 continued 3. According to Han Solo made illegal modifications to make the Millennium Falcon do 25,000 light-years per day. At this speed, how much time does Han save by making the Kessel Run in 12 parsecs rather than 18 parsecs? Some key landmarks to look for in judging this item: parsecs is 39 light years parsecs is 58.5 light years 3. Han can make the 12-parsec run in days or hours or minutes 4. Han can make the 18-parsec run in days or hours or minutes 5. Hans saves minutes or 67.4 seconds 12

13 Part 7 That s No Moon In Episode IV, A New Hope, the following scene occurs: [The Falcon jumps out of hyperspace into a dense field of rocks and debris.] Han Solo: What the? Luke Skywalker: What's going on? Han Solo: Our position is correct, except, no Alderaan. Luke Skywalker: What do you mean? Where is it? Han Solo: Thats what I'm trying to tell you, kid. It ain't there. It's been totally blown away. Luke Skywalker: What!? How? Ben Kenobi: [grimly] Destroyed. By the Empire. Han Solo: The entire star fleet couldn't destroy the whole planet. It'd take a thousand ships with more fire power than I've there's another ship coming in! Luke Skywalker: Maybe they know what happened. [A TIE fighter flies over] Ben Kenobi: It's an Imperial fighter. Luke Skywalker: It followed us! Ben Kenobi: No, it's a short-range fighter. Han Solo: There aren't any bases around here. Where did it come from? Luke Skywalker: It sure is leaving in a big hurry. If they identify us, we're in big trouble. Han Solo: Not if I can help it. Chewie, jam its transmissions. Ben Kenobi: It'd be as well to let it go. It's too far out of range. Han Solo: Not for long. Ben Kenobi: A fighter that size couldn't get this deep into space on its own. Luke Skywalker: He must have gotten lost, been part of a convoy or something. Han Solo: Well, he ain't going to be around long enough to tell anyone about us. Luke Skywalker: Look at him. He's headed for that small moon. Han Solo: I think I can get him before he gets there. He's almost in range. Ben Kenobi: [with sudden realization] That's no moon. It's a space station. Han Solo: It's too big to be a space station. Luke Skywalker: I have a very bad feeling about this. Ben Kenobi: Turn the ship around. Han Solo: Yeah. I think you're right How big is the Death Star battle station? One source indicates that it has a diameter of 140 kilometers. Earth s moon has a diameter of 3474 kilometers. How many Death Star battle stations could fit inside of Earth s moon? Assuming both the Death Star and the moon to be spherical: 4 3 π(1737km)3 4 15, 279 Death Star battle stations will fit inside Earth s moon. 3 π(70km)3 2. In Episode VI, Return of the Jedi, a new Death Star is found to be under construction. Suppose that the Empire wanted to make it so large that it was 1/100 the volume of the Earth s moon. What would the diameter of the new Death Star need to be to accomplish this goal? Round to the nearest tenth of a kilometer. Assuming a spherical moon and Death Star, the new Death Star would need to have a diameter of km. 13

14 Part 8 Box Office Sales The graph shows the daily cumulative domestic box office (think total ticket sales) for all of the Star Wars movie. Use this graph to estimate responses to the following items. 1. Which Star Wars episode shows the greatest rate of change in daily cumulative domestic box office? What is this rate of change? Star Wars Episode VII shows the greatest rate of change from Day 0 to about Day 16. This rate of change is about $720,000 per 16 days or about $45,000 per day. This is where we see the greatest increase in daily cumulative domestic box office relative to the change in the number of days. 2. The daily cumulative domestic box office for which episode could best be modeled with a linear function? Why? Create a linear function model for the graph of the cumulative domestic box office for this episode of Star Wars. Star Wars Episode V and VI are most linear as seen by their more consistent rate of increase of cumulative domestic box office. Episode V: CDBO = 1.64d (in millions of dollars) Episode VI: CDBO = 2.29d (in millions of dollars) 14