2016 AP Calculus AB Free Response #5

Allen Tsao The STEM Coach
13 Sept 201808:34
EducationalLearning
32 Likes 10 Comments

TLDRIn this engaging video, Alan from Bottle Stem continues his series on AP Calculus 2016 free response questions, focusing on question number five. He tackles the problem of finding the average value of the radius of a funnel with a height of ten inches and a radius given by the function R(H) = โˆš(3 + H^2), where R and H are measured in inches. Alan demonstrates the process of integration to find the average radius and then moves on to calculate the volume of the funnel by summing the volumes of infinitesimally thin discs. He also addresses a rate of change problem, determining the rate at which the height of liquid in the funnel is decreasing when the liquid's height is three inches and the radius is decreasing at a rate of 150 inches per second. Despite the complexity and the lack of a calculator, Alan manages to solve the problems step by step, providing a detailed walkthrough of the mathematical process. He concludes by sharing his thoughts on the difficulty of the question and offers additional help through his Twitch or Discord channels for those interested in math and physics.

Takeaways
  • ๐Ÿ“š The video is a continuation of AP Calculus 2016 free response questions, focusing on question number five.
  • ๐Ÿ“ The problem involves a funnel with a height of ten inches, and the radius of its cross-sections varies with height H according to a given formula.
  • โญ• The average value of the funnel's radius is calculated by integrating the radius function from 0 to 10 and then dividing by the interval width (10).
  • ๐Ÿงฎ The integral to find the average radius involves the function R(H) = 1/(23 + H^2), which is integrated and simplified to find the average value.
  • ๐Ÿš€ The volume of the funnel is determined by summing the volumes of infinitesimally thin discs, each with a variable radius and thickness.
  • ๐Ÿ”ข The volume calculation requires integrating the area of the disc (ฯ€R^2) times its thickness (DH) over the height of the funnel.
  • โณ A rate of change problem is presented, asking for the rate at which the height of the liquid in the funnel is changing with respect to time when the liquid's height is three inches and the radius is decreasing at a rate of 150 inches per second.
  • ๐Ÿ”„ To solve for the rate of change in height, the derivative of the radius with respect to time (dr/dt) is used, applying the chain rule to find dh/dt.
  • ๐Ÿ“‰ The rate of change of the liquid's height with respect to time (dh/dt) is found to be negative two-thirds inches per second at the given conditions.
  • ๐ŸŽฏ The presenter successfully calculates the average radius, volume, and rate of change, indicating a good understanding of calculus concepts.
  • ๐Ÿ“ˆ The presenter notes that the 2016 AP Calculus questions felt more tedious numerically compared to the 2017-2018 ones.
  • ๐Ÿ’ป The video concludes with an offer for free homework help on Twitch or Discord for those interested in learning more about math and physics.
Q & A
  • What is the formula given for the radius of the funnel at a certain height H?

    -The radius of the funnel at height H is given by the formula R = 1/(23 + H^2), where R and H are in inches.

  • How is the average value of the radius of the funnel calculated?

    -The average value of the radius is calculated by integrating the radius function R(H) with respect to H from 0 to 10, and then dividing by the width of the interval, which is 10 in this case.

  • What is the final result of the integral for the average radius of the funnel?

    -The final result of the integral is approximately 30 + 1000/200, which simplifies to 30 + 5, giving a total of 35 inches.

  • How is the volume of the funnel calculated?

    -The volume of the funnel is calculated by slicing it into thin discs and summing up the volumes of these discs. The volume of each disc is found by multiplying the area of the disc (ฯ€R^2) by its thickness (dH).

  • What is the expression for the volume of a single disc in the funnel?

    -The volume of a single disc is given by the expression ฯ€ * (1/(23 + H^2))^2 * dH.

  • What is the final expression for the volume of the funnel?

    -The final expression for the volume of the funnel is ฯ€/400 * (9H + 2H^3 + 1/5 * H^5) evaluated from H = 0 to H = 10.

  • What is the result of the volume calculation for the funnel?

    -The result of the volume calculation is ฯ€/40 * 22090 cubic inches.

  • What is the rate of change of the radius of the liquid surface with respect to time at H equals 3 inches?

    -The rate of change of the radius with respect to time at H equals 3 inches is given as -1/5 inch per second.

  • How is the rate of change in the height of the liquid with respect to time calculated?

    -The rate of change in the height of the liquid with respect to time (dh/dt) is calculated using the chain rule, where dr/dt is known and H is the height at which the rate is being calculated.

  • What is the rate of change in the height of the liquid with respect to time at H equals 3 inches?

    -At H equals 3 inches, the rate of change in the height of the liquid with respect to time (dh/dt) is -2/3 inches per second.

  • What does Alan offer for those who need help with homework or want to learn about math and physics?

    -Alan offers free homework help on platforms like Twitch or Discord for those who have homework questions or want to learn about different parts of math and physics.

  • What is the final parting message from Alan to the viewers?

    -Alan thanks the viewers for watching, asks them to leave a comment, like, or subscribe, and informs them that he will see them in the final AP Calculus free response for the exam.

Outlines
00:00
๐Ÿ“š Calculus Problem: Funnel's Average Radius and Volume

In the first paragraph, Alan discusses an AP Calculus 2016 free response question focused on a funnel with a height of ten inches. The funnel's radius at any height H is given by a specific function. Alan calculates the average value of the funnel's radius by integrating the radius function from 0 to 10 and dividing by the interval width. He then moves on to calculate the volume of the funnel by slicing it into discs and summing their volumes. This involves integrating a function of the radius squared over the same interval. Alan mentions the complexity of the math involved and the absence of a calculator, which makes the process more challenging.

05:04
๐Ÿ’ง Rate of Change in Liquid Height within a Funnel

The second paragraph deals with a rate of change problem concerning the liquid inside the funnel. At a specific instant when the liquid's height is three inches, the radius of the liquid surface is decreasing at a rate of 150 inches per second. Alan uses the given radius function and the rate of change of the radius to find the rate of change of the liquid's height with respect to time. He applies the chain rule to differentiate the equation and solves for the rate of change in height (dh/dt), which he calculates to be negative two-thirds inches per second. Finally, Alan discusses the scoring guideline for the problem and expresses satisfaction with his solution, noting the algebra involved was more complex than expected.

Mindmap
Keywords
๐Ÿ’กAP Calculus
AP Calculus is a high school course offered by the College Board that covers topics in calculus, including limits, derivatives, integrals, and series. In the video, Alan is discussing free response questions from the 2016 AP Calculus exam, which is a significant part of the theme as it sets the context for the mathematical problems being solved.
๐Ÿ’กFree Response Questions
Free response questions are a type of question found on certain standardized tests, including AP exams, where students must write out their answers in prose or mathematical form. Alan is working through these types of questions in the video, which require a deeper understanding and application of concepts as opposed to multiple-choice questions.
๐Ÿ’กIntegral
An integral in calculus is a mathematical concept that represents the area under a curve defined by a function. In the video, Alan uses integration to find the average value of the radius of a funnel, which is a key part of solving the problem presented.
๐Ÿ’กVolume
Volume refers to the amount of space occupied by a three-dimensional object. Alan calculates the volume of a funnel in the video by summing up the volumes of infinitesimally thin discs that make up the funnel, which is a practical application of integral calculus.
๐Ÿ’กRate of Change
The rate of change, often denoted as 'dr/dt' for radius with respect to time, is a measure of how quickly a quantity is changing over time. Alan uses this concept to determine how fast the height of a liquid in the funnel is changing when the radius of the liquid surface is decreasing at a certain rate.
๐Ÿ’กChain Rule
The chain rule is a fundamental theorem in calculus used to compute the derivative of a composite function. In the script, Alan applies the chain rule to find the rate of change of the height of the liquid with respect to time, which is a crucial step in solving the rate of change problem.
๐Ÿ’กDisc Method
The disc method is a technique used to calculate the volume of a solid of revolution by slicing the solid into discs and summing their volumes. Alan uses this method to find the volume of the funnel, which is a central part of the video's content.
๐Ÿ’กAverage Value
The average value of a function over an interval is a single value that represents the mean of the function's values over that interval. Alan calculates the average value of the radius of the funnel, which is an important concept in understanding the properties of the funnel's cross-sections.
๐Ÿ’กFunnel
A funnel is a tool used to pour liquids or fine-grained materials into containers with small openings. In the video, the funnel serves as the geometric object of interest, with its dimensions and the contained liquid's properties being the focus of the mathematical problems.
๐Ÿ’กCross-Sections
Cross-sections in geometry refer to the shapes that are created by cutting through an object with a plane. The video discusses the circular cross-sections of a funnel at varying heights, which is essential for understanding the geometry involved in the volume calculation.
๐Ÿ’กTwitch
Twitch is a live streaming platform often used for gaming but also for various other content, including educational streams. Alan mentions offering free homework help on Twitch, indicating an additional platform where viewers can engage with him for further learning.
๐Ÿ’กDiscord
Discord is a communication platform that allows for text, voice, and video communication, often used by communities and for educational purposes. Alan offers help on Discord, suggesting it as another avenue where he interacts with students and provides assistance with homework.
Highlights

Alan is discussing AP Calculus 2016 free response question number five.

The problem involves finding the average value of the radius of a funnel with a height of ten inches.

The radius of the funnel at height H is given by the function R(H) = โˆš(3 + H^2).

The average value of the radius is calculated by integrating R(H) from 0 to 10 and dividing by the interval width.

Alan simplifies the integral to 1/200 * (3H + 1/3 * H^3) evaluated from 0 to 10.

The final result for the average radius is approximately 30.1667 inches.

Next, Alan calculates the volume of the funnel by slicing it into discs and summing their volumes.

The volume of a single disc is given by ฯ€ * R^2 * dH, where R is the radius and dH is the disc thickness.

The volume integral is set up as ฯ€/400 * (9 + 6H^2 + H^4) * dH from 0 to 10.

After evaluating the integral, the volume of the funnel is found to be (ฯ€/4) * 20909 cubic inches.

A rate of change problem is addressed, where the liquid in the funnel is draining, and the radius decreases at a rate of 150 inches per second when H = 3.

The derivative dr/dt is given as -1/5, and Alan uses it to find dh/dt using the chain rule.

The rate of change in the height of the liquid with respect to time, dh/dt, is calculated to be -2/3 inches per second.

Alan provides the scoring guideline for the question, which matches his calculations.

He comments on the difficulty level of the question, noting it to be more tedious than previous ones.

Alan offers free homework help on Twitch or Discord for those interested in learning math and physics.

He encourages viewers to comment, like, or subscribe for more educational content.

Transcripts
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