The Amusement Park

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 t9, there
are no people in the park.
10

• How many people have entered the park by 5 PM
  (t17)? Round answer to the nearest whole number.
  
• The price of admission to the park is 15 until
  500 PM (t17). After 500 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 900 a.m. and stays open until
  1100 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 900 a.m. and stays open until
  1100 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 900 a.m. and stays open until
  1100 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 900 a.m. and stays open until
  1100 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 900 a.m. and stays open until
  1100 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 900 a.m. and stays open until
  1100 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 900 AM (t 9), when
  the park opens, until 1100 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
  500 PM (t 17). After 500 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 900 AM (t 9), when
  the park opens, until 1100 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
  500 PM (t 17). After 500 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 900 AM (t 9), when
  the park opens, until 1100 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
  500 PM (t 17). After 500 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 900 AM (t 9), when
  the park opens, until 1100 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
  500 PM (t 17). After 500 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 900 AM (t 9), when
  the park opens, until 1100 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
  500 PM (t 17). After 500 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
  500 PM (t 17). After 500 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
  500 PM (t 17). After 500 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
  500 PM (t 17). After 500 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
  500 PM (t 17). After 500 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
  (t12)? 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
  500 PM (t17). After 500 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
  (t12)? 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
  500 PM (t17). After 500 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
  (t12)? 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
  500 PM (t17). After 500 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
  (t12)? 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
  500 PM (t17). After 500 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 -gt 3 pts
2 pts
Find value of H'(17).
2 pts
3 pts -gt 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 t9, there
are no people in the park.
34

• How many people have entered the park by 5 PM
  (t17)? Round your answer to the nearest whole
  number.
• The price of admission to the park is 15 until
  500 PM (t17). After 500 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
  (t17)? 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 500 PM (t17). After 500 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
grade percentage Range of score Average score
5 18.0 69 7.39
4 23.0 46 5.00
3 26.3 14 2.43
BC average 4.90, SD 2.81
grade percentage Range of score Average score
5 43.3 69 7.46
4 16.3 56 5.30
3 21.6 25 3.40
41

1. When is the rate at which people are entering the
   park greatest?
2. At what time is the total number of people in the
   park increasing at the greatest rate?
3. What is the greatest number of people in the park
   at any time?
4. How many people are in the park when it closes at
   1100 PM?
5. What is the average amount of time each person
   spends in the park?

42
Start with easier question What if everyone
stays until 1100 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 1100 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 1100 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 1100 PM.
46
Adjust for those who leave before 1100 PM.
Total number who enter the park is
47
Adjust for those who leave before 1100 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/talk
s