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The Amusement Park Problem

This presentation discusses a mathematical problem involving the entrance and exit rates of an amusement park. It explains how to calculate the number of people in the park at a given time and the revenue collected from ticket sales. The problem is solved using mathematical functions and equations.

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The Amusement Park Problem

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  1. The Amusement Park Problem David M. Bressoud Macalester College, St. Paul, MN Bellingham, WA, May 20, 2006 This PowerPoint will be available at www.macalester.edu/~bressoud/talks

  2. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split

  3. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts

  4. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts Fall 2004: review 3rd drafts

  5. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts Fall 2004: review 3rd drafts Spring 2005: finalize the free-response problems

  6. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts Fall 2004: review 3rd drafts Spring 2005: finalize the free-response problems Summer 2005: review final version, especially in light of 2005 exam; review multiple choice exam

  7. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts Fall 2004: review 3rd drafts Spring 2005: finalize the free-response problems Summer 2005: review final version, especially in light of 2005 exam; review multiple choice exam Fall 2005: check printers proofs of free response questions, finalize the multiple choice problems

  8. The Schedule for the 2006 exam Spring 2004: select problems for operational, alternate, and overseas exams; for each exam there are 3 common problems, 3 AB only, 2 BC only, 1 AB/BC split Summer 2004: review 2nd drafts Fall 2004: review 3rd drafts Spring 2005: finalize the free-response problems Summer 2005: review final version, especially in light of 2005 exam; review multiple choice exam Fall 2005: check printers proofs of free response questions, finalize the multiple choice problems November-December 2005: chief reader and committee chair check printers proofs of multiple choice exam

  9. 2002 AB2/BC2 The rate at which people enter an amusement park on a given day is modeled by the function E defined by The rate at which people leave the same amusement park is modeled by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for , the hours during which the park is open. At time t=9, there are no people in the park.

  10. How many people have entered the park by 5 PM (t=17)? Round answer to the nearest whole number. The price of admission to the park is $15 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. Let for . The value of H(17) to the nearest whole number is 3725. Find the value of H'(17) and explain the meaning of H(17) and H'(17) in the context of the park. At what time t, for , does the model predict that the number of people in the park is a maximum?

  11. 2000: AB4 Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of gallons per minute, for minutes. At time t = 0, the tank contains 30 gallons of water. How many gallons of water leak out of the tank from time t = 0 to t = 3 minutes? How many gallons of water are in the tank at time t = 3 minutes? Write an expression for A(t), the total number of gallons of water in the tank at time t. At what time t, for , is the amount of water in the tank a maximum? Justify your answer.

  12. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar.

  13. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar.

  14. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar.

  15. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar.

  16. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar. Bad question - no calculus

  17. The Source On the average day during the summer months the rate that people enter a particular amusement park at any time t (in hours) is estimated by the function E(t). The rate that they leave is estimated by the function L(t). Both E(t) and L(t) are given in hundreds of people per hour. The park opens at 9:00 a.m. and stays open until 11:00 p.m. Graph y = E(t) and y = L(t) on the axes provided. To the nearest hundred how many people have entered the park by noon? Let N(t) represent the number of hundreds of people in the park at any time t. Write an expression for N(t). At what time (to the nearest minute) is the largest number of people in the park? If each ticket costs $20, how much revenue is collected on an average day? Express your answer to the nearest dollar. Same question

  18. January 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9:00 AM (t = 9), when the park opens, until 11:00 PM (t = 23), when the park closes. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole person. Let N(t) represent the number of people in the park at time t. Write an expression for N(t) . At what time t is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar.

  19. January 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9:00 AM (t = 9), when the park opens, until 11:00 PM (t = 23), when the park closes. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole person. Let N(t) represent the number of people in the park at time t. Write an expression for N(t) . At what time t is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar.

  20. January 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9:00 AM (t = 9), when the park opens, until 11:00 PM (t = 23), when the park closes. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole person. Let N(t) represent the number of people in the park at time t. Write an expression for N(t) . At what time t is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar. Better to use an interval: 9 ≤ t ≤ 23

  21. January 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9:00 AM (t = 9), when the park opens, until 11:00 PM (t = 23), when the park closes. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole person. Let N(t) represent the number of people in the park at time t. Write an expression for N(t) . At what time t is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar.

  22. January 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9:00 AM (t = 9), when the park opens, until 11:00 PM (t = 23), when the park closes. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole person. Let N(t) represent the number of people in the park at time t. Write an expression for N(t) . At what time t is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar.

  23. March 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for the hours during which the park is open. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole number. Let N(t) represent the number of people in the park at time t, for . Write an expression for N(t) . At what time t, for is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar. 2 pts 2 pts 2 pts 3 pts

  24. March 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for the hours during which the park is open. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole number. Let N(t) represent the number of people in the park at time t, for . Write an expression for N(t) . At what time t, for is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar. 2 pts 2 pts 2 pts 3 pts e) Find the maximum number of people in the park.

  25. March 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for the hours during which the park is open. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole number. Let N(t) represent the number of people in the park at time t, for . Write an expression for N(t) . At what time t, for is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar. 2 pts 2 pts 2 pts 3 pts e) At what time is the number of people in the park rising fastest, and when is the rate at which people are entering the greatest?

  26. March 2001 The rate at which people enter an amusement park on a given day is approximated by the function E defined by . The rate at which people leave the same amusement park is approximated by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for the hours during which the park is open. How many people have entered the park by noon (t = 12)? Round your answer to the nearest whole number. Let N(t) represent the number of people in the park at time t, for . Write an expression for N(t) . At what time t, for is the greatest number of people in the park? The price of admission to the park is $20 until 5:00 PM (t = 17). After 5:00 PM, the price of admission to the park is $15. Approximately how many dollars are collected from admission to the park on a given day? Give your answer to the nearest dollar. 2 pts 2 pts 2 pts 3 pts e) What is the average amount of time a person spends in the park?

  27. July 2001 The rate at which people enter an amusement park on a given day is modeled by the function E defined by The rate at which people leave the same amusement park is modeled by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for , the hours during which the park is open.

  28. July 2001 The rate at which people enter an amusement park on a given day is modeled by the function E defined by The rate at which people leave the same amusement park is modeled by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for , the hours during which the park is open. Put denominators inside parentheses!

  29. How many people have entered the park by noon (t=12)? Round answer to the nearest whole number. Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park. At what time t, for , is the number of people in the park greatest? The price of admission to the park is $20 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $15. According to this model, how many dollars are collected from admissions to the park on the given day? Round your answer to the nearest dollar.

  30. How many people have entered the park by noon (t=12)? Round answer to the nearest whole number. Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park. At what time t, for , is the number of people in the park greatest? The price of admission to the park is $20 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $15. According to this model, how many dollars are collected from admissions to the park on the given day? Round your answer to the nearest dollar.

  31. How many people have entered the park by noon (t=12)? Round answer to the nearest whole number. Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park. At what time t, for , is the number of people in the park greatest? The price of admission to the park is $20 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $15. According to this model, how many dollars are collected from admissions to the park on the given day? Round your answer to the nearest dollar. 2 pts 2 pts 2 pts 3 pts

  32. How many people have entered the park by noon (t=12)? Round answer to the nearest whole number. Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park. At what time t, for , is the number of people in the park greatest? The price of admission to the park is $20 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $15. According to this model, how many dollars are collected from admissions to the park on the given day? Round your answer to the nearest dollar. 2 pts -> 3 pts 2 pts Find value of H'(17). 2 pts 3 pts -> 1 pt

  33. October 2001 The rate at which people enter an amusement park on a given day is modeled by the function E defined by The rate at which people leave the same amusement park is modeled by the function L defined by Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for , the hours during which the park is open. At time t=9, there are no people in the park.

  34. How many people have entered the park by 5 PM (t=17)? Round your answer to the nearest whole number. The price of admission to the park is $15 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. Let for . The value of H(17) to the nearest whole number is 3725. Find the value of H'(17) and explain the meaning of H(17) and H'(17) in the context of the amusement park. At what time t, for , does the model predict that the number of people in the park is a maximum?

  35. How many people have entered the park by 5 PM (t=17)? Round answer to the nearest whole number. 3 points: 2 pts for set-up (1 for correct limits, 1 for correct integrand), 1 pt for answer

  36. b) The price of admission to the park is $15 until 5:00 PM (t=17). After 5:00 PM, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. 1 point: set-up only

  37. c) Let for . The value of H(17) to the nearest whole number is 3725. Find the value of H'(17) and explain the meaning of H(17) and H'(17) in the context of the park. 3 points: 1 pt for H'(17) and 1 pt for each explanation H(17) is the number of people in the park at time t = 17. H'(17) is the rate at which the number of people in the park is changing. The number of people in the park is decreasing at a rate of approximately 380 people per hour. interpretation implication

  38. Ambiguous • The rate of the number of people at 5 PM • The rate at which people enter/leave (enter and leave, enter or leave) at 5 PM • The rate at which people stay at 5 PM • This compares the rates at which people enter and leave at 5 PM • Acceptable • The rate of change of the number of people at 5 PM • The rate at which the number of people is changing at 5 PM • The net rate of change of the number of people at 5 PM • The difference between rate of entering and rate of leaving at 5 PM • Unacceptable • Any reference to average rate of change • Any reference to something occurring over an interval of time • The number of people leaving the park at 5 PM • The rate which people leave the park at 5 PM

  39. d) At what time t, for , does the model predict that the number of people in the park is a maximum? 2 points: 1 pt for recognizing that this occurs when E(t) –L(t) = 0, 1 pt for finding the answer: t = 15.794 or t = 15.795 hours after midnight.

  40. AB average: 3.13, SD = 2.82 BC average: 4.90, SD = 2.81

  41. When is the rate at which people are entering the park greatest? At what time is the total number of people in the park increasing at the greatest rate? What is the greatest number of people in the park at any time? How many people are in the park when it closes at 11:00 PM? What is the average amount of time each person spends in the park?

  42. Start with easier question: What if everyone stays until 11:00 PM? If someone enters at time t, they spend 23 – t hours in the park. How many people enter at time t?

  43. Start with easier question: What if everyone stays until 11:00 PM? If someone enters at time t, they spend 23 – t hours in the park. How many people enter at time t? Number of people who enter between time t and time is

  44. Start with easier question: What if everyone stays until 11:00 PM? If someone enters at time t, they spend 23 – t hours in the park. How many people enter at time t? Number of people who enter between time t and time is

  45. Adjust for those who leave before 11:00 PM.

  46. Adjust for those who leave before 11:00 PM. Total number who enter the park is

  47. Adjust for those who leave before 11:00 PM. Total number who enter the park is Answer is

  48. APCentral at apcentral.collegeboard.com SIGMAA TAHSM (Special Interest Group of the MAA, Teaching Advanced High School Mathematics) at www.maa.org/SIGMAA/tahsm/ This PowerPoint presentation at www.macalester.edu/~bressoud/talks

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