2009 AP Calculus AB Free Response #2
TLDRIn this engaging video, Alan from Bottle Stem Coach guides viewers through a problem from the 2009 AP Calculus exam. He tackles the problem of determining the number of people in an auditorium when a rock concert begins, using the function R(T) to model the rate of entry. Alan demonstrates how to calculate the integral from 0 to 2 to find the total number of people. He also explores the maximum rate of entry by finding the derivative of R(T) and checking endpoints. Additionally, Alan calculates the total wait time for all attendees and the average wait time per person, using the function W and its derivative. The video is a practical demonstration of calculus concepts, including the Fundamental Theorem of Calculus, and offers valuable insights for those learning to apply calculus in real-world scenarios.
Takeaways
- 📘 The rate at which people enter the auditorium for a rock concert is modeled by the function R(T), measured in people per hour.
- 🕒 At time T=0, no one is in the auditorium when the doors open, and the concert begins at T=2 when the doors close.
- 🧮 To find out how many people are in the auditorium when the concert begins, integrate R(T) from 0 to 2.
- 📊 Alan suggests practicing doing the integral by hand or using a calculator like a TI, which is helpful for those learning the process.
- 🔢 The integral calculation results in 980 people being in the auditorium when the concert begins.
- 📈 To find the time when the rate of people entering the auditorium is maximum, take the derivative of R(T) and find where it equals zero.
- 🔍 After calculating, the maximum rate occurs at T = 1.358, which is within the interval [0, 2], and is not an endpoint.
- 🕒 The total wait time for all people in the auditorium is found by integrating the derivative of the wait time function W from 0 to 2.
- ⏱️ The average wait time for a person is the total wait time divided by the number of people, which is calculated to be approximately 0.776 hours.
- 👥 Alan emphasizes the importance of considering all people who enter the auditorium after the doors open for calculating the average wait time.
- 💡 The video is educational, providing step-by-step calculations and reasoning, which is beneficial for viewers learning calculus.
- 📢 Alan invites viewers to engage with the content by leaving comments, liking, or subscribing for more educational content.
Q & A
What is the function R(t) used for in the context of the video?
-The function R(t) is used to model the rate at which people enter the auditorium for a rock concert, measured in people per hour.
At what time does the concert begin?
-The concert begins at t equals 2.
How many people are in the auditorium when the concert begins?
-The number of people in the auditorium when the concert begins is found by integrating R(t) from 0 to 2, which results in 980 people.
What does Alan suggest to practice the integral calculation?
-Alan suggests practicing the integral calculation by hand and also by entering it into a calculator, specifically a TI calculator, to refresh his skills and to show viewers how it can be done.
How does Alan determine the time at which the rate of people entering the auditorium is maximum?
-Alan determines the time at which the rate is maximum by finding the local maximums of the derivative of R(t), which is done by setting the derivative equal to zero and solving for t.
What is the total wait time for all the people in the auditorium?
-The total wait time for all the people in the auditorium is found by integrating the derivative of the wait time function W from 0 to 2, which results in 387.5 hours.
How does Alan calculate the average wait time for a person?
-Alan calculates the average wait time for a person by dividing the total wait time by the number of people who entered the auditorium.
What is the total wait time for those who entered the auditorium after time T equals 1?
-The total wait time for those who entered after time T equals 1 is found by integrating (2 - T) * R(T) from 1 to 2, which results in 387.5 hours.
What is the model used to calculate the average wait time for all people who entered after the doors opened?
-The model used to calculate the average wait time is the total wait time divided by the number of people, where the total wait time is the integral from 0 to 2 of W'(T) dT.
What was the final calculated average wait time for a person?
-The final calculated average wait time for a person is approximately 0.77 hours.
What does Alan offer to help viewers with their homework?
-Alan offers free homework help on platforms like Twitch and Discord.
How can viewers catch up on more content and get in touch with Alan?
-Viewers can catch up on more content and get in touch with Alan by leaving a comment, liking the video, subscribing to the channel, and following the links provided in the video description.
Outlines
📚 Calculus Exam Question: Auditorium Entry Rate
In this paragraph, Alan, along with Bottle Stem and Coach, is working on a question from the 2009 AP Calculus exam. The question involves modeling the rate at which people enter an auditorium for a rock concert using the function R(T), measured in people per hour. The challenge is to find out how many people are in the auditorium when the concert begins at T equals 2. Alan suggests using a calculator to solve the integral from 0 to 2 of R(T) DT, and demonstrates how to do this on a TI calculator. He also discusses finding the time when the rate of people entering the auditorium is at its maximum by taking the derivative of R(T) and looking for critical points. The maximum rate is determined by evaluating R(T) at the endpoints and the critical point found from the derivative.
🕒 Total Wait Time and Average Wait Time Calculation
The second paragraph focuses on calculating the total wait time for all people in the auditorium before the concert begins. The function W is introduced to represent the total wait time, and its derivative is used to find the wait time for people who enter after a certain time T. Alan uses the Fundamental Theorem of Calculus to compute the integral from T=1 to T=2 of the given function, which represents the total wait time for those who entered after time T=1. To find the average wait time, Alan calculates the total wait time by integrating from 0 to 2 of W'(T) DT and then divides this by the number of people, which is given as 980. Despite a minor calculation error, Alan corrects himself and provides the final calculation for the average wait time. The paragraph concludes with a summary of the results and an invitation for viewers to engage with the content and seek further help through offered platforms.
Mindmap
Keywords
💡AP Calculus Exam
💡Function R(T)
💡Integral
💡Derivative
💡TI Calculator
💡Concert
💡Total Wait Time
💡Average Wait Time
💡Fundamental Theorem of Calculus
💡Modeling
💡Free Homework Help
Highlights
Alan with Bottle Stem continues the AP Calculus exam series.
The rate at which people enter an auditorium for a rock concert is modeled by a function R(T).
R(T) is measured in people per hour, with no one in the auditorium at time T=0 when the doors open.
The doors close, and the concert begins at T=2.
The task is to find out how many people are in the auditorium when the concert begins by integrating R(T) from 0 to 2.
Alan suggests practicing the integral by hand or using a calculator, specifically a TI calculator.
He demonstrates entering the integral into a calculator and provides a window for the function's graph.
The integral calculation results in approximately 980 people in the auditorium when the concert begins.
Alan then finds the time when the rate of people entering the auditorium is maximum by taking the derivative of R(T).
The derivative of R(T) is calculated and set to zero to find the local maximums.
The maximum rate occurs at T=1.358, which is less than T=2, so it is a valid candidate for the maximum rate.
Alan checks the endpoints T=0 and T=2 to ensure the maximum rate is correctly identified.
The total wait time for all people in the auditorium is found by integrating the derivative of the wait time function W from T=0 to T=2.
The average wait time for a person is calculated by dividing the total wait time by the number of people.
The total wait time is calculated as 387.5 hours, and the average wait time per person is 0.7754 hours.
Alan makes a minor error in his calculations but corrects it to provide the accurate total wait time and average wait time.
He encourages viewers to practice similar calculations and provides additional resources for homework help on Twitch and Discord.
Alan concludes the video by thanking viewers and inviting them to comment, like, subscribe, and engage with the content.
Transcripts
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