2008 AP Calculus AB Free Response #3
TLDRIn this educational video, Alan from Bottle Stem Coaching tackles a 2008 AP Calculus question involving a leaking oil pipeline and the resulting oil slick in a lake. The oil slick forms a right circular cylinder with its volume increasing at a constant rate of 2000 cubic centimeters per minute. Alan explains how to calculate the rate of change of the height of the oil slick using calculus, given the initial conditions and rates of change for radius and height. He then addresses a scenario where a recovery device starts removing oil at a rate proportional to the square root of time, and determines the time at which the oil slick reaches its maximum volume. Finally, Alan calculates the total volume of oil present when the recovery device begins its work, using an integral to account for the oil added and removed over time. The video concludes with a brief review of the solutions and an invitation for viewers to engage with the content and seek further assistance through offered platforms.
Takeaways
- π The video covers a solution to question 3 from the 2008 AP Calculus exam, which involves using a graphing calculator.
- π¨ The problem scenario describes oil leaking from a pipeline into a lake, forming an oil slick that takes the shape of a right circular cylinder with changing dimensions.
- π The volume (V) of the oil slick is defined by the formula (V = Ο r^2 h), where (r) is radius and (h) is height.
- π The rate of change of the volume ( dV/dt ) is given as 2000 cubic centimeters per minute.
- π‘ The question asks to find the rate at which the height (h) of the oil slick changes when (r = 100) cm and (h = 0.5) cm.
- π Using the product rule to differentiate (V) with respect to time, the formula derived is ( dV/dt = Ο (2rh (dr/dt) + r^2 (dh/dt)) ).
- π‘ By isolating (dh/dt) and substituting given values, the change rate of height was calculated as approximately 0.0387 cm/min.
- π© Part B of the question introduces a recovery device that affects the oil volume by removing oil at a rate ( R(t) = 400 βt ) cubic centimeters per minute.
- π The maximum volume of the oil slick occurs when the net rate of change of the volume becomes zero, calculated to be at (t = 25) minutes.
- π The final part involves calculating the total volume of the oil at (t = 25) minutes, considering the initial leaked volume and the net change due to leakage and recovery.
Q & A
What is the rate at which the volume of the oil slick increases?
-The volume of the oil slick increases at a constant rate of 2000 cubic centimeters per minute.
What shape does the oil slick take?
-The oil slick takes the form of a right circular cylinder with both the radius and height changing with time.
What is the formula for the volume of a right circular cylinder?
-The volume V of a right circular cylinder is given by the equation V = ΟR^2H, where R is the radius and H is the height.
What is the rate of change of the radius of the oil slick at the instant when the radius is 100 centimeters?
-The rate of change of the radius (dr/dt) is 2.5 centimeters per minute.
At what rate is the height of the oil slick changing when the radius is 100 centimeters and the height is 0.5 centimeters?
-The height is changing at a rate of 0.0387 centimeters per minute.
When does the oil slick reach its maximum volume?
-The oil slick reaches its maximum volume at 25 minutes after the recovery device begins removing oil.
What is the total volume of oil that has leaked by the time the recovery device begins removing oil?
-By the time the recovery device begins removing oil, 60,000 cubic centimeters of oil had already leaked.
What is the rate at which oil is removed by the recovery device?
-The rate at which oil is removed by the recovery device is given by the function R(t), where t is the time in minutes since the device began working.
What is the integral expression used to find the volume of the oil slick at the time found in Part B?
-The volume of the oil slick at the time found in Part B is given by the integral expression 60,000 + β«(2000 - 400βt) dt from 0 to 25.
What is the final volume of the oil slick after the recovery device has been working for 25 minutes?
-The final volume of the oil slick after 25 minutes is 60,000 + 225β25 cubic centimeters.
What is the significance of the rate of change of the volume being zero in determining the maximum volume of the oil slick?
-The rate of change of the volume being zero signifies that the volume of the oil slick is no longer increasing, indicating that it has reached its maximum volume.
How does the script help in understanding the application of calculus in real-world scenarios?
-The script demonstrates the application of calculus in modeling and solving real-world problems, such as calculating the rate of change of a physical quantity (volume of an oil slick) and determining when it reaches a maximum.
Outlines
π Analyzing Oil Leak Dynamics and Calculating Rate of Change
Alan, from Bottle Stem Coach, tackles a question related to a leaking oil pipeline. The problem involves a graphing calculator question where oil is leaking from a pipeline onto the surface of a lake, forming an oil slick in the shape of a right circular cylinder. Alan breaks down the given equation for the volume of the cylinder and derives an expression for the rate of change of the height of the oil slick over time. He then proceeds to solve for the rate of change of the height at a specific instant when the radius and height are given values. Through differentiation and substitution, he arrives at the rate of change of the height as 0.0387 centimeters per minute.
π Determining Time for Oil Slick to Reach Maximum Volume
Continuing the discussion, Alan explores the scenario further to find the time when the oil slick reaches its maximum volume. He reasons that the maximum volume occurs when the rate of change of volume is zero. Setting up an equation with the rates of oil leakage and removal, he solves for the time at which the volume reaches its maximum. By equating the rates of oil gain and removal, he obtains the time T as 25 minutes, indicating when the oil slick reaches its maximum volume.
Mindmap
Keywords
π‘Graphing Calculator
π‘AP Calculus
π‘Volume
π‘Rate of Change
π‘Right Circular Cylinder
π‘Leak
π‘Derivative
π‘Product Rule
π‘Recovery Device
π‘Maximum Volume
π‘Integral
Highlights
Alan introduces the 2008 AP Calculus question focusing on graphing calculator problems.
The problem involves an oil leak from a pipeline forming a right circular cylinder oil slick on a lake surface.
The volume of the oil slick increases at a constant rate of 2000 cubic centimeters per minute.
The volume V of the oil slick is given by the equation V = ΟR^2H, where R is the radius and H is the height.
The rate of change of the volume (dV/dt) is provided as 2000 cubic centimeters per minute.
At a specific instant, the radius is 100 centimeters, and the height is 0.5 centimeters, with the radius increasing at 2.5 cm/min.
Using the product rule, Alan derives the equation for the rate of change of the height (dH/dt).
The equation for dH/dt is solved to find the rate of height change at the given instant.
The calculated rate of height change is 0.0387 centimeters per minute.
A recovery device arrives and begins removing oil at a rate R(t), where t is the time in minutes.
The oil continues to leak at a rate of 2000 cubic centimeters per minute.
The time T when the oil slick reaches its maximum volume is determined by setting the rate of volume change to zero.
The maximum volume is reached at T = 25 minutes.
By the time the recovery device began, 60,000 cubic centimeters of oil had already leaked.
The volume of oil at the time the recovery device began is calculated using an integral from zero to 25 minutes.
The integral calculation accounts for the oil added and removed, resulting in a volume of 60,000 + 20,000Ο cubic centimeters.
Alan concludes the video by summarizing the solution and encouraging viewers to comment, like, or subscribe.
Transcripts
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