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 y2 y1 C 2 r A bh m 2 x 2 x 1 Circumference of a circle Area of a triangle Slope 12 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.

5 Each part of this Team Project will focus on some aspect of the 2014 Olympic Games. Have fun! 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. 2. Suppose that the scores with the asterisks (*) were the scores that were randomly eliminated. Find the average score for Mao Asada. 3. 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.

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 the 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. Rank Name of Skater Speed (MPH)

7 Distance from Start (meters) Greatest (1) (2) (3) (4) Least (5) 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) 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.

8 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. 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?

9 6. 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, Case I or II? Explain how you know. 7. According to the graph, which skater, A or B, won the race? Explain. 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 Maximum 2. Create a five-number summary for the total medal count for the Germany. Minimum Lower Quartile Median Upper Quartile Maximum

10 3. On the graph below, create a box-and-whisker plot for both data sets showing the medal counts for the USA and Germany. Place one plot above the other and label which is which. 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.

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

12 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. 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.

13 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. Part 6 Continues 2. Measure the angle of attack in Figure 1 using a protractor. Record that angle measure here. 3. Measure the angle of attack in Figure 2 using a protractor. Record that angle measure here.

14 Measure of Angle A 4. 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. 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

15 $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. 2. 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. 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

16 Olympic cause in order to cover the $50 billion cost? Explain your results. 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.