2013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

Transcription

1 013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

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 l w A r Area of a rectangle Perimeter of a rectangle Area of a circle b dr s r d 1 C r A bh d a Circumference of a circle Area of a triangle Arc Length a b c 580 feet = 1 mile s r 360 Pythagorean Theorem b dy s 1 dx dx.54 centimeters = 1 inch a Arc Length h 4.9t v0t h0 h 16t v0t h0 1 kilogram =. pounds 1 meter = inches 1 gigabyte = 1000 megabytes 1 mile = 1609 meters 1 gallon = 3.8 liters 1 square mile = 640 acres s r 1 cu. ft. of water = 7.48 gallons 1 ml = 1 cu. cm. 4 3 V r h V Area of Base height V r 3 Volume of cylinder Volume Volume of a sphere Lateral SA = r h Lateral surface area of cylinder b b 4ac x a Quadratic Formula tan sin cos

3 TEAM PROJECT Level I 013 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. We encourage the use of a graphing calculator. Part 1: Background The death spiral is a move performed by paired figure skaters as shown in the picture. The proctor of the Team Project will be showing a video showing different styles of the death spiral performed by skaters. In this activity, you will analyze certain quantities that can be measured while skaters perform the death spiral. As the video shows, there are many different styles of the death spiral. However, for this project, you will focus on the following death spiral situation. Suppose it is known that AB is 9 feet and that this distance does not vary. While the death spiral is performed, the man s height ( AC ) changes as he squats down while the woman slides away from the man and distance CB changes. Suppose at the time that AC measures 5 feet, the woman is sliding away at the rate of 1 foot per second. A C B 3

4 Part : Computing Distances and Speeds 1. Using the information in Part 1, determine the length of CB. Let x = length of CB, y = length of AC, and z = length of AB. x y z x 5 81 x 56 x 56 feet. Examine the series of screenshots below showing the position of the skaters in one second intervals. It shows one pair of skaters engaged in the death spiral. The first image captures, approximately, the moment when the pair is first fully engaged in the death spiral. Notice that the woman s right foot sweeps out a nearly circular path. Calculate the average speed traveled by her right foot during the death spiral. The woman s right food completes one revolution in seconds. The radius of the circle swept out is Therefore, her foot travelled a distance of C feet in seconds. Therefore, the speed is computed as follows: 56 feet speed 3.51 feet per second seconds 56 feet. 4

5 Part continues 3. The Colossus Ferris wheel located at Six Flags Amusement Park in St. Louis, MO is shown in the picture. Specific facts about the Colossus are provided. Work to develop a thorough response to the following question: It has been reported that the Colossus turns at a rate of 1½ revolution per minute (en.wikipedia.org). Is the top speed of the Colossus more than or less than the speed of the skater s foot calculated in #? As part of your work, determine the speed that would likely be used in the chart (note the question mark). Provide a mathematical justification for your response. That is, the mathematics that you present should clearly communicate the rationale for your answer to the question. C 165 feet At a rate of 1½ revolution per minute, the Colossus will travel? revolutions per minute 165 feet per revolution 47.5 feet per minute feet per minute feet 1 minute 1.96 feet per second 1 minute 60 seconds feet 60 minutes 1 mile 8.8 MPH 1 minute 1 hour 580 feet The skater travels about 1.8 times faster than the Colossus. 5

6 Part 3: Analyzing the Change in the Man s Height Work to develop a thorough response to the following question: Given the quantities and rates provided in Part 1, is the rate at which the man lowers his body greater than or less than the rate at which the lady slides horizontally away from the man? Provide a mathematical justification for your response. That is, the mathematics that you present should clearly communicate the rationale for your answer to the question. Let x = length of CB, y = length of AC, and z = length of AB. x y z dx dy dz x y z dt dt dt dy dt dy dt 5 5 The man is lowering his body at a rate of about feet per second while the woman is sliding away at a rate of 1 foot per second. The man is lowering his body at a greater rate than the rate at which the woman is sliding away (horizontally) from the man. 6

7 Part 4: Analyzing the Change in an Angle Again using the quantities and rates provided previously, work to develop a thorough response to the following question: How fast does the measure of ABC change? Provide a mathematical justification for your response. That is, the mathematics that you present should clearly communicate the rationale for your answer to the question. The measure of Let = measure of ABC 56 arccos 9 x cos z d 1 dx sin dt z dt 56 d 1 sin arccos 1 9 dt 9 5 d 1 9 dt 9 d 1 dt 5 ABC is decreasing at a rate of 1 radian per second. 5 7

8 Part 5: The Angle Function Continue to use the given quantities and rates. If the distance AC decreases as the man lowers his body, does the measure of ABC increase or decrease? How do you know? Draw a graph illustrating the relationship between the man s height and the measure of ABC. Write a short description of how one might read/interpret this graph. That is, how does the graph make sense in the context of this situation? Let = measure of ABC y sin z y y arcsin arcsin z 9 m (radians) (feet) The graph shows that as the measure of length AC decreases, the measure of ABC also decreases. 8

9 Part 6: The Length of the Death Spiral The reason that this figure skating move is called the death spiral is because of the path created by the skates of the woman skater as she enters into and completes the move. At first, the woman s skates are near to the man s skates and the distance between their skates increases as the woman revolves and begins the move. The graph shown is a model of this path and is the polar graph of r 1.45 where r is measured in feet. 1. In radians, construct an argument for the interval through which changes to create the woman skater s path. That is, to model the skater s path from the beginning of the death spiral move until she reaches the full radius of the circle described in Part, explain what interval of values of best model this movement. Answers will vary. Look for a viable argument. For example, consider the graph on. This could model the path of the skates entering into the death spiral move. This interval puts the woman s skates about.5 feet at the start of the move and at 9 feet as she moves into the full circular path (at.. Determine the length of the spiral in this model using the interval you created in #1 above. b dr s r d d a (1.45 ) feet d 9