2014 Excellence in Mathematics Contest Team Project

Transcription

1 2014 Excellence in Mathematics Contest Team Project School Name: Group Members: 2014 Scott Adamson

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3 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 2 A r 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 inches = 1 foot 5280 feet = 1 mile 3 feet = 1 yard 16 ounces = 1 pound 2.54 centimeters 1 inch 100 = $1 1 kilogram 2.2 pounds 1 ton = 2000 pounds 1 gigabyte = 1000 megabytes 1 mile = 1609 meters 1 gallon 3.8 liters 1 square mile = 640 acres 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

4 TEAM PROJECT 2014 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 The Winter Olympics The Winter Olympic Games were first held in 1924 in Chamonix, France. Except for the 1940 and 1944 games, the Winter Games have been held every four years from in the same year as the Summer Olympic Games. In 1986, the International Olympic Committee decided to place the Summer and Winter Games on separate four-year cycles so that an Olympic Games would be held every two years. Therefore, the next Winter Games after 1992 were held in 1994 in Lillehammer, Norway. The Winter Games continue to be held every four years and will be held in 2014 in Sochi, Russia. The following excerpt, from provides background information related to the host city for the 2014 Winter Games. The 2014 Olympic Winter Games will be the first time that the Russian Federation will have hosted the Winter Games; the Soviet Union hosted the 1980 Summer Games in Moscow. The host city Sochi has a population of 400,000 people and is situated in Krasnodar, which is the third largest region in Russia. The Games will be organised in two clusters: a coastal cluster for ice events in Sochi, and a mountain cluster located in the Krasnaya Polyana Mountains. This will make it one of the most compact Games ever, with around 30 minutes travel time from the coastal to mountain cluster. The Sochi Olympic Park will be built along the Black Sea coast in the Imeretinskaya Valley, where all the ice venues such as the Bolshoi Ice Palace, the Maly Ice Palace, the Olympic Oval, the Sochi Olympic Skating Centre, the Olympic Curling Centre, the Central Stadium, the Main Olympic Village and the International Broadcast Centre and Main Press Centre, will be built anew for the 2014 Games. The Park will ensure a very compact concept with an average distance of 6km between the Olympic Village and the other coastal venues. The mountain cluster in Krasnaya Polyana will be home to all the skiing and sliding sports. The mountain concept is again a very compact one with only an average distance of 4km between the mountain sub-village and the venues. There will also be a sub-media centre in the mountain cluster. Each part of this Team Project will focus on some aspect of the 2014 Olympic Games. Have fun!

5 Part 2 Figure Skating Each participant in the women s figure skating event is observed by nine judges. Each judge awards points for different elements of the skater s program. Ultimately, each judge gives a final score for each performance. However, only 5 of the judge s scores will count in the final tabulations. Two judge s scores are randomly eliminated. The highest score is eliminated. The lowest score is eliminated. This leaves 5 scores from 5 of the judges. These scores are averaged. Consider the following scores from the 2010 Winter Games held in Vancouver, British Columbia, Canada. They represent the scores of each of the 9 judges for one aspect of the free skating program for the gold and silver medalists for the event. Kim Yu Na (Korea) * * Mao Asada (Japan) 8.50* * Suppose that the scores with the asterisks (*) were the scores that were randomly eliminated. Find the average score for Kim Yu Na Suppose that the scores with the asterisks (*) were the scores that were randomly eliminated. Find the average score for Mao Asada Write a detailed, mathematical explanation of what the average score for Kim Yu Na means. To be clear, make sure you explain what the average score means and NOT how it was computed. The average score of 9.05 takes the total points (45.25) and equally distributes the points among the 5 remaining judges so that it is as if each judge gave the same score of Hypothetically, if each judge awarded a score of 9.05, the skater would have earned the same total points, 42.25, as actually were awarded.

6 Part 3 Speed Skating Another Winter Olympic event that was made most popular by Apollo Ono (to the far left in the picture) of the United States is speed skating. There are several different speed skating events held on both a short track and a long track. The short track oval has a circumference of meters. The long track oval has a circumference of 400 meters. At the short track, races are held at five different distances, 500 meters, 1000 meters, 1500 meters, 5000 meters, and 10,000 meters. The table shows the Olympic records in each of the different short track distances. Distance Olympic Record Holder Stopwatch Time* Year and Place 500 meters Casey FitzRandolph ( ) Salt Lake City 1000 meters Gerard van Velde ( ) 1: Salt Lake City 1500 meters Derek Parra ( ) 1: Salt Lake City 5000 meters Sven Kramer ( ) 6: Vancouver meters Lee Seung-hoon ( ) 12: Vancouver *Stopwatch time is of form MINUTES:SECOND for example 6:14.60 means 6 minutes, seconds 1. By finding the average speed of each in miles per hour, rank the skaters in order from greatest to least average speed. Fill in your rankings in the table below. Round each speed to the hundredths place. 500 m 3600 seconds miles MPH seconds 1 hour 1 meter 1000 m 3600 seconds miles MPH seconds 1 hour 1 meter 1500 m 3600 seconds miles MPH seconds 1 hour 1 meter 5000 m 3600 seconds miles MPH seconds 1 hour 1 meter m 3600 seconds miles MPH seconds 1 hour 1 meter Rank Name of Skater Speed (MPH) Greatest (1) Gerard van Velde (1000 m) (2) Casey FitzRandolph (500 m) (3) Derek Parra (1500 m) (4) Sven Kramer (5000 m) Least (5) Lee Seung-hoon (10000 m) 28.73

7 Distance from Start (meters) Part 3 Continues 2. Suppose Casey FitzRandolph skated at his average speed throughout the entire race. Create a graph of Casey s distance (in meters) from the starting line as a function of time (in seconds) for the duration of the race. Write a brief explanation of why you made the graph the way you made it. Be as precise as possible. Elapsed Time (seconds) Assuming a constant speed of MPH (14.53 meters per second), the graph of distance from start (in meters) versus elapsed time (in seconds) is linear. If his speed was constant, he would cover the same amount of distance in any particular given interval of time during any part of the race. 3. If possible, write a function formula for the graph you created in #2 above. Explain how you were able to create this function formula. If it is not possible, explain why. The function formula is D twhere D represents the distance from start (in meters) and t represents the elapsed time in the race. At an average speed of meters per second, we multiply this by the time elapsed to find the total distance traveled. 4. Suppose Casey FitzRandolph was able to compete in the 10,000 meter race and was able to maintain his average speed found in #1. What would his finish time have been? Would he have set a record? Comment on whether or not Casey would be able to complete the 10,000 meters in this way t t t seconds At an average speed of meters per second, Casey would have completed the 10,000 meter race in seconds or 11: This would be an Olympic record. It is unlikely that Casey would be able to keep this pace for the entire 10,000 meter race.

8 Part 3 Continues 5. The following graph shows the progression of average world record speeds for the different race lengths. Clearly describe the details of the most dramatic increase in the record speed that can be observed in the graph. When did it occur? Which race and gender? How do you know? The women s 5,000 meter race experienced the greatest increase in average world record speed in approximately We see that the average speed had been holding steady for about 20 or 25 years at approximately 21 MPH and then increased to about 24 MPH around The website shares a video where two different 5000 meter racing scenarios are presented. In one case (Case I), it is proposed that a racer starts out fast but then tires out and slows down as he finishes the race. In the other case (Case II), it is proposed that the racer starts out slower but then finishes the race relatively quickly. The following graph is used to represent these situations. Which graph, A or B, corresponds with which scenario, I or II? Explain how you know. Case I B Case II A A good response will clearly describe the relationship between distance and time in justifying their conclusion. A mediocre response will only point to the steepness of the graph with no attempt to describe the covarying nature of the quantities. 7. According to the graph, which skater, A or B, won the race? According to the graph, the race was a tie. Each skater traveled the required 5000 meters in the same amount of time thus making the race a tie.

9 Part 4 Gold Medal Count Since 1976, the United States and Germany have been among the top countries that have earned medals (gold, silver, and bronze) in the Winter Olympic Games. The following table shows the total number of medals earned by each country since the 1976 games. Year 1976 Innsbruck 1980 Lake Placid 1984 Sarajevo 1988 Calgary 1992 Albertville 1994 Lillehammer 1998 Nagano 2002 Salt Lake City 2006 Turin 2010 Vancouver USA Germany* *East Germany until Create a five-number summary for the total medal count for the USA. Minimum Lower Quartile Median Upper Quartile or or 25 Maximum 37 NOTE: Q1 and Q3 answers can vary depending on method. 2. Create a five-number summary for the total medal count for the Germany. Minimum Lower Quartile Median Upper Quartile Maximum or or NOTE: Q1 and Q3 answers can vary depending on method. 3. On the graph below, create a box-and-whisker plot for both the data showing the medal counts for the USA and Germany.

10 Part 4 Continues 4. Write a summary statement (several sentences) that compares and contrasts what the box-and-whisker plots for the United States and Germany show regarding the medal counts in the Winter Olympic Games in the given time period. Look for some of the following important features: The relative shortness of the box for Germany shows a consistent number of high medal counts compared to the relative longness of the box for the US which shows that the number of medals earned is more spread out. The median medal count for Germany is much higher than for the United States. In fact, the median medal count for Germany is nearly the same as the upper quartile medal count for the US. The lower quartile for the US is much more compact than the lower quartile for Germany. This speaks to the relative spread of the data in this lower quartile. The data in the upper quartiles are nearly equally spread when comparing the two nations.

11 Part 5 The Skeleton Event Skeleton is an event at the Winter Olympic Games where participants ride a sled, head first, down the bobsled track. They are allowed a running start but then must jump on the sled, belly down, and navigate the course. The course in Sochi has a length of 1,814 meters and participants will experience a vertical drop of meters. The course is designed to have 19 curves that must successfully be navigated by the participants. The goal is to finish the course in the fastest time. There has been some controversy surrounding the design of the course in Sochi. It is required that the local event team submits the downhill grade to Olympic officials to ensure safety. It was reported that the downhill grade of seven sights have been submitted but Olympic officials have rejected them all due to high downhill grade on the track. While the downhill grade varies at different places along the course, your task is to use the given quantities and report an overall downhill grade for the course. In the space below, clearly communicate the mathematical thinking involved in your response. You may use words, mathematical symbols, computations, and/or pictures as needed to best communicate your strategy. Answers may vary. Here is a strategy Suppose we straighten out the track (hypothetically). Using the given quantities, we can create the following picture (not to scale). Use the Pythagorean Theorem to compute the length of the horizontal side meters 1814 meters meters L L L L meters Therefore, the grade is: or about 7.3%

12 Part 6 Ice Hockey An understanding of fundamental geometry may be helpful in the sport of ice hockey. Particularly, hockey players can benefit by understanding the idea of angle measure. You will need to use the protractor that has been provided to respond to the items in this part of the project. First, study the images and descriptions below. Then answer the questions. The following images show different locations of the puck that the offensive team is trying to shoot into the opposing team s net. The goaltender would do his or her best to prevent the puck from going into the net. Consider the angle of attack (in yellow) which is the angle formed where the puck is the vertex of the angle and the sides of the angle extend to the corners of the goalie s crease (in blue). We will call the angle that the puck makes with the center of the back of the crease and the horizontal red line the observed angle. In Figure 1, the observed angle is 90 o. In Figure 2, the observed angle is 45 o. 90 o 45 o Figure 1: The puck is located directly in front of the net. Figure 2: The puck is located at a 45 o angle as shown. 1. Think about and discuss with your team the mathematical definition of angle. As clearly and precisely as possible, write a definition of angle. That is, if you were to describe the idea of angle to someone who does not know what one is, what would you say? Include in your description both what an angle is and how an angle is measured. Answers will vary. Here is the general idea that we are looking for an angle is an object and we can measure a particular attribute of that object. An angle is the union of two rays that have a common endpoint. The measure of the angle s openness can be described as being the fraction of a circle s circumference that the angle cuts out (assuming that the vertex of the angle is located at the center of the circle).

13 Measure of Angle A Part 6 Continues 2. Measure the angle of attack in Figure 1 using a protractor. Record that angle measure here Measure the angle of attack in Figure 2 using a protractor. Record that angle measure here Let the measure of the angle of attack be represented with the variable A. Let the measure of the observed angle be represented by the variable O. Create a rough graph of the measure of angle A as a function of the measure of angle O. Measure of Angle O 5. Write a description of the graph you created in #4. That is, what information does it provide relative to the situation? Be as clear and precise as possible. Answers will vary. As we judge, look for sophistication both in the description and in the graph provided. For example, does the response precisely indicate a maximum? Is the graph made up (incorrectly) of linear segments or is it curved? Either way, does the response provide an explanation using rate of change ideas? For example, a good response would recognize that the angle of attack only changed by about 4 degrees for the first increase of 45 degrees in the observed angle. In the last 45 degrees (taking the puck to the red line), the angle would change from about 11 degrees to 0 degrees a much more dramatic change. Thus, the graph should be concave down. A good response might indicate that hockey players might take as a strategy the idea of working to keep the puck within the first 45 degrees on either side of the front of the net. Moving the puck within this range does not change the angle of attack as much as it does in the lower 45 degrees on either side of the net.

14 Part 7 The Economic Impact of the Olympics in Sochi Consider the following excerpt from Sochi has already set one record. At an estimated cost of $50 billion, these will be the most expensive games in history. When Russia placed its bid in 2007 it proposed to spend $12 billion, already more than any other country. Within a year the budget had been replaced by a seven-year plan to develop Sochi as a mountain resort. Most of the money is coming from the public purse or from state-owned banks. Allison Stewart, of the SAID Business School at Oxford, says that Olympics tend to have cost overruns of about 180% on average. For Sochi the overrun is now 500%%. But Russia made clear that money was not an issue, says Ms Stewart. She also notes that relations between the government and construction companies appear closer in Sochi than in other games. Large construction projects often have a side-effect of corruption. But in Russia corruption is not a side-effect: it is a product almost as important as the sporting event itself. 1. Note the blocked out portion of the article. It reports the cost overrun is now %. Based on the information in the article, what number should go in the blank? Show and/or explain the mathematical work needed to answer this question % The unit of currency in Sochi is the Russian Ruble. Using Google, the following information was discovered (retrieved 12/28/13). Use this information to determine the cost, in Rubles, of hosting the Winter Olympic Games assuming that the estimated $50 billion amount is correct. $50,000,000,000 1,612,903,226,000 Russian Rubles

15 Part 7 Continues 3. According to sochimagazine.com, the average monthly salary in Sochi is 32,100 rubles. The population of Sochi is 343,334 (google.com). Considering that not all of the population is working (e.g. kids, elderly, retired), we will assume that 40% of the population is working (similar to the US). To gain perspective on the cost of hosting the Olympics, determine how many month s salary would the workers in Sochi hypothetically have to donate to the Olympic cause in order to cover the $50 billion cost? Explain your results , ,334 workers in Sochi 1, 612,903, 226, 000 Russian Rubles 50, 246, 207 monthly salaries 32,100 rubles per person per month 50,246, , A total of about 50,246,207monthly salaries or about 366 monthly salaries from each worker in Sochi would have to be donated in order to cover the cost of hosting the Olympic Games. 4. According to the Bureau of Labor Statistics, the million workers in the United States as of May 2011 hold jobs ranging from fast-food workers requiring no education and earning barely above minimum wage to CEOs with MBAs who earned more than nine times that amount. The population of the US is about million people. The population of Salt Lake City, UT is 189,314 people. The average monthly salary in Salt Lake City is $4,333. How many monthly salaries from the people in Salt Lake City would hypothetically have to be donated to the Olympic cause in order to cover the $50 billion cost? Assume that the percentage of workers in Salt Lake City is the same as for the US. Explain your results. $50,000,000,000 11,539,349 monthly salaries $4,333 per person per month The percentage of workers in the US (and therefore Salt Lake City) is 128,200, % 313,900, ,314 77,619 workers in Salt Lake City 11,539, , 619 A total of about 11,539,349monthly salaries or about 149 monthly salaries from each worker in Salt Lake City would have to be donated in order to cover the cost of hosting the Olympic Games.