2010 AP Calculus AB Free Response #3
TLDRIn the video, Alan from Bothell Stem Coach tackles AP Calculus AB 2010 Free-Response Question 5, focusing on the non-calculator portion. The problem involves a popular amusement park ride with an initial queue of 700 people. The ride operates for 8 hours, accepting passengers at a rate of 800 people per hour. People arrive at varying rates throughout the day, and Alan uses the concept of the area under the curve to estimate the number of people who arrive between 0 and 3 hours. He discusses the increasing queue size and determines that the line is longest at 3 hours when the arrival rate equals the ride's capacity. Alan also attempts to find the time when the queue disappears by solving an integral expression. However, he makes a mistake by initially forgetting the initial 700 people in the queue, which he later corrects. The video concludes with a reminder that Alan offers free homework help on Twitch and Discord.
Takeaways
- 🎢 The amusement park ride starts with 700 people in line and operates for 8 hours.
- ⏰ The ride accepts passengers at a rate of 800 people per hour.
- 📈 The rate at which people arrive forms a graph where time is measured in hours.
- 🔢 To find out how many people arrived between 0 and 3 hours, you calculate the area under the graph, which can be approximated as the sum of two trapezoids.
- 📉 The number of people waiting in line increases between 2 and 3 hours because the arrival rate is higher than the ride's capacity.
- 🆙 The line for the ride is longest when the arrival rate equals the ride's capacity, which happens at 3 hours.
- 🧮 The integral from 0 to t of the difference between the arrival rate and 800 people per hour gives the number of people in line at time t.
- ✅ To find the maximum line length, set the derivative of the integral equal to zero, which occurs when the arrival rate equals 800 people per hour.
- 🕒 The line is longest at approximately 3 hours into the operation.
- 🚫 The presenter made a mistake by initially forgetting to include the initial 700 people in the line, which affected subsequent calculations.
- 🗣️ The presenter offers free homework help on Twitch and Discord for further assistance.
Q & A
How many people were initially in line for the amusement park ride?
-There were initially 700 people in line for the amusement park ride.
What is the rate at which the ride accepts passengers per hour?
-The ride accepts passengers at a rate of 800 people per hour.
How many hours does the amusement park operate before closing?
-The amusement park operates for 8 hours before closing.
What is the method used to calculate the number of people who arrived at the ride between zero and three hours?
-The method used is to calculate the area under the curve graphically, which is divided into two trapezoids.
What is the total number of people that arrived at the ride between zero and three hours?
-The total number of people that arrived at the ride between zero and three hours is 3,200.
Is the number of people waiting in line to get on the ride increasing or decreasing between 2 and 3 hours?
-The number of people waiting in line is increasing between 2 and 3 hours because the rate of people arriving (R of T) is greater than 800 people per hour during that time.
At what time is the line for the ride the longest?
-The line for the ride is the longest at time T equals 3 hours.
What is the integral expression used to find the number of people in line at a given time?
-The integral expression used is the integral from 0 to t of R of T minus 800 dt.
What is the earliest time T at which there's no longer a line for the ride?
-The earliest time T at which there's no longer a line for the ride is when the integral from 0 to t of R of T minus 800 dt equals zero.
How many people were on the line at 3 hours?
-At 3 hours, there were 3,200 people who had arrived minus 2,400 people who got onto the ride, which equals 800 people on the line.
What was the mistake made in the calculation of the number of people on the line at 3 hours?
-The mistake was not including the initial 700 people already in line when the park opened, which should have been added to the final calculation.
What additional resources does Alan offer for homework help?
-Alan offers free homework help on Twitch and Discord.
Outlines
🎢 Amusement Park Ride Queue Analysis
In this paragraph, Alan from Bothell Stem Coach discusses an AP Calculus problem set in 2010. The problem involves calculating the number of people in line for a popular amusement park ride that starts with 700 people and operates for 8 hours. The rate at which people join the line varies, and the ride can accommodate 800 people per hour. Alan uses the concept of the area under a graph to represent the number of people arriving and leaving the queue. He breaks down the problem into two trapezoids and calculates the area to find out how many people arrived between 0 and 3 hours. He also addresses whether the line is increasing or decreasing and determines the time when the line is the longest. Finally, Alan identifies a mistake in his calculation due to an oversight of the initial 700 people in line and corrects it.
🧮 Correcting the Queue Calculation Error
This paragraph focuses on correcting the error made in the previous calculation. Alan realizes that he forgot to include the initial 700 people already in line at the start of the ride's operation. He acknowledges the mistake and corrects the calculation by adding 700 to the total number of people who arrived. Alan then discusses the integral expression to find the earliest time when there is no longer a line for the ride, indicating that the solution to this expression gives the time when the number of people in line is zero. He concludes by summarizing the corrected calculations and encourages viewers to engage with the content through comments, likes, or subscriptions. Alan also offers free homework help on Twitch and Discord and signs off, promising to see viewers in the next video.
Mindmap
Keywords
💡AP Calculus AB
💡Non-calculator portion
💡Amusement park ride
💡Rate
💡Integral
💡Trapezoids
💡Fundamental Theorem of Calculus
💡Maximum
💡Derivative
💡Area under the curve
💡Graphical method
Highlights
Alan is discussing AP Calculus AB 2010 exam questions.
The focus is on the non-calculator portion of the exam.
A scenario involving an amusement park ride with 700 people in line is presented.
The rate at which people move on to the ride is 800 people per hour.
The problem involves calculating the number of people who arrive at the ride between 0 and 3 hours.
Alan visualizes the area under the graph representing the rate of people's arrival as two trapezoids.
The area under the curve is calculated graphically without a calculator.
The total number of people who arrived within the first three hours is 3,200.
The line for the ride is longest when the number of people arriving equals the ride's capacity of 800 people per hour.
The maximum line length occurs at T equals 3 hours.
Alan makes a mistake by forgetting to include the initial 700 people in the line, which affects the calculation.
The correct number of people in line at the maximum point should include the initial 700 people.
Alan provides a method to find the earliest time when there's no longer a line for the ride by solving an integral expression.
The final calculation reveals that at 3 hours, 2,424 people have boarded the ride.
Alan offers free homework help on Twitch and Discord for further assistance.
The video concludes with an invitation to engage with the content through comments, likes, or subscriptions.
Transcripts
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