David M. Bressoud

Macalester College, St. Paul, MN

Bellingham, WA, May 20, 2006

This PowerPoint will be available at www.macalester.edu/~bressoud/talks

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

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

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

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

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

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

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

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.

E t( ) =15600

t2 −24t+160( )

L t( ) =9890

t2 −38t+ 370( )

9 ≤t≤23

a) How many people have entered the park by 5 PM (t=17)? Round answer to the nearest whole number.

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.

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.

d) At what time t, for , does the model predict that the number of people in the park is a maximum?

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

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.

t +10 ≤t≤120

0 ≤t≤120

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

Bad question - no calculus

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.

a) Graph y = E(t) and y = L(t) on the axes provided.b) To the nearest hundred how many people have entered the park by

noon?c) Let N(t) represent the number of hundreds of people in the park at

any time t. Write an expression for N(t).d) At what time (to the nearest minute) is the largest number of people

in the park?e) If each ticket costs $20, how much revenue is collected on an average

day? Express your answer to the nearest dollar.

E t( ) =9.17

x−1( ) / 4( )2+1

L t( ) =8

x−8( ) / 3( )2+1

Same question

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.

E t( ) =15000

t2 −24t+160L t( ) =

9200t2 −38t+ 370

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.

E t( ) =15000

t2 −24t+160L t( ) =

9200t2 −38t+ 370

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.

E t( ) =15000

t2 −24t+160L t( ) =

9200t2 −38t+ 370

Better to use an interval: 9 ≤ t ≤ 23

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.

E t( ) =15000

t2 −24t+160L t( ) =

9200t2 −38t+ 370

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.

E t( ) =15000

t2 −24t+160L t( ) =

9200t2 −38t+ 370

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.

E t( ) =15300

t2 −24t+160L t( ) =

9200t2 −38t+ 370

9 ≤t≤23,

9 ≤t≤23

9 ≤t≤23,

2 pts

2 pts2 pts

3 pts

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.

E t( ) =15300

t2 −24t+160L t( ) =

9200t2 −38t+ 370

9 ≤t≤23,

9 ≤t≤23

9 ≤t≤23,

2 pts

2 pts2 pts

3 pts

e) Find the maximum number of people in the park.

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.

E t( ) =15300

t2 −24t+160L t( ) =

9200t2 −38t+ 370

9 ≤t≤23,

9 ≤t≤23

9 ≤t≤23,

2 pts

2 pts2 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?

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.

E t( ) =15300

t2 −24t+160L t( ) =

9200t2 −38t+ 370

9 ≤t≤23,

9 ≤t≤23

9 ≤t≤23,

2 pts

2 pts2 pts

3 pts

e) What is the average amount of time a person spends in the park?

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.

E t( ) =15300

t2 −24t+160

L t( ) =9200

t2 −38t+ 370

9 ≤t≤23

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.

E t( ) =15300

t2 −24t+160

L t( ) =9200

t2 −38t+ 370

9 ≤t≤23

Put denominators inside parentheses!

a) How many people have entered the park by noon (t=12)? Round answer to the nearest whole number.

b) Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park.

c) At what time t, for , is the number of people in the park greatest?

d) 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.

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

a) How many people have entered the park by noon (t=12)? Round answer to the nearest whole number.

b) Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park.

c) At what time t, for , is the number of people in the park greatest?

d) 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.

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

a) How many people have entered the park by noon (t=12)? Round answer to the nearest whole number.

b) Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park.

c) At what time t, for , is the number of people in the park greatest?

d) 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.

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

2 pts

2 pts

2 pts

3 pts

a) How many people have entered the park by noon (t=12)? Round answer to the nearest whole number.

b) Let for . Explain the meaning of H(12) and H'(12) in the context of the amusement park.

c) At what time t, for , is the number of people in the park greatest?

d) 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.

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

2 pts -> 3 pts

2 pts

2 pts

3 pts -> 1 pt

Find value of H'(17).

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.

E t( ) =15600

t2 −24t+160( )

L t( ) =9890

t2 −38t+ 370( )

9 ≤t≤23

a) How many people have entered the park by 5 PM (t=17)? Round your answer to the nearest whole number.

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.

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 amusement park.

d) At what time t, for , does the model predict that the number of people in the park is a maximum?

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

9 ≤t≤23

a) 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

E t( )dt=6004 people9

17

∫

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

$15 × E t( )dt9

17

∫ +$11× E t( )dt17

23

∫ =$104048

or

$15 ×6004 +$11×1271=$104041

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( ) =E 17( )−L 17( ) =−380.281 people/hour

H t( ) = E x( )−L x( )( )dx9

t

∫ 9 ≤t≤23

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

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

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.

9 ≤t≤23

AB average: 3.13, SD = 2.82

grade percentage Range of score Average score

5 18.0% 6–9 7.39

4 23.0% 4–6 5.00

3 26.3% 1–4 2.43

BC average: 4.90, SD = 2.81

grade percentage Range of score Average score

5 43.3% 6–9 7.46

4 16.3% 5–6 5.30

3 21.6% 2–5 3.40

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 11:00 PM?

5. What is the average amount of time each person spends in the park?

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?

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?

E t( )×ΔtNumber of people who enter between time t and time is t + Δt

E t( )×Δt× 23−t( ) people hours spent in park

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?

E t( )×ΔtNumber of people who enter between time t and time is t + Δt

E t( )×Δt× 23−t( ) people hours spent in park

E ti( )∑ 23−ti( )Δt → E t( ) 23−t( )dt people hours9

23

∫

Adjust for those who leave before 11:00 PM.

E t( ) 23−t( )dt9

23

∫ − L t( ) 23−t( )dt=30382.933 people hours9

23

∫

Adjust for those who leave before 11:00 PM.

E t( ) 23−t( )dt9

23

∫ − L t( ) 23−t( )dt=30382.933 people hours9

23

∫

Total number who enter the park is

E t( )dt9

23

∫ =7275.553

Adjust for those who leave before 11:00 PM.

E t( ) 23−t( )dt9

23

∫ − L t( ) 23−t( )dt=30382.933 people hours9

23

∫

Total number who enter the park is

E t( )dt9

23

∫ =7275.553

Answer is

30382.933 people hours

7275.553 people=4.176 hours = 4 hours, 11 minutes

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