2011 Calculus AB free response #5a | AP Calculus AB | Khan Academy

Khan Academy
9 Sept 201106:59
EducationalLearning
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TLDRThe video script discusses a mathematical problem involving an increasing function W that models the total amount of solid waste in a landfill. It explains the use of a tangent line at the initial point to approximate the waste amount after 3 months (1/4 year) in 2010. By applying the given differential equation, the derivative of W at t=0 is calculated to determine the slope of the tangent line, leading to an estimated 1,411 tons of waste after the specified time period.

Takeaways
  • 📈 The problem discusses a model of solid waste accumulation at a landfill, with the function W representing the total waste amount in tons.
  • 🕒 The model is based on a differential equation predicting waste growth over a 20-year period, starting from 2010.
  • 🔢 The differential equation is given as dW/dt = 1/25 * (W - 300), where W is in tons and t is in years.
  • 🏗️ The landfill starts with 1,400 tons of waste at t=0, and the function W is increasing over time.
  • 📊 Part A of the problem asks to approximate the waste amount at the end of the first 3 months (1/4 year) using the tangent line at t=0.
  • 🤔 The key to solving Part A is understanding the slope of the tangent line at t=0, which is the derivative of W at that point.
  • 🧮 By evaluating the derivative at t=0, we find the slope to be 44 (1/25 * 1400 - 300).
  • 📈 Using the slope and the initial condition, we create a linear approximation to predict the waste amount after 1/4 year.
  • 📌 The equation of the approximation is W_approx = 44 * t + 1400, where t is in years.
  • 🔍 Plugging in t=1/4 into the approximation equation, we get an estimated waste amount of 1,411 tons after 3 months.
  • ⚠️ The approximation may not be exact but serves as a reasonable estimate based on the given model and information.
Q & A

  • What was the initial amount of solid waste in the landfill at the start of 2010?

    -The initial amount of solid waste in the landfill at the start of 2010 was 1,400 tons.

  • How is the increasing function W defined in the context of the landfill?

    -The increasing function W models the total amount of solid waste stored at the landfill, and it is expected to satisfy a specific differential equation over the next 20 years.

  • What is the differential equation that the function W is estimated to satisfy?

    -The differential equation that W is estimated to satisfy is dW/dt = 1/25 * (W - 300), where W is measured in tons and t is measured in years from the start of 2010.

  • How is the derivative of W with respect to time related to the function itself?

    -The derivative of W with respect to time represents the rate of change or the slope of the tangent line to the graph of W at any given time, which indicates how the amount of waste is increasing or decreasing.

  • What is the significance of using the tangent line to the graph of W at t equals 0?

    -Using the tangent line to the graph of W at t equals 0 allows for the approximation of the future amount of solid waste in the landfill by using the initial slope of the function W, which represents the starting growth rate of the waste.

  • How is the slope of the tangent line at t equals 0 calculated?

    -The slope of the tangent line at t equals 0 is calculated by taking the derivative of W with respect to t at that point, which is 1/25 * (W at t=0 - 300), and substituting the known initial condition of W (1,400 tons) into the equation.

  • What is the approximate amount of solid waste in the landfill at the end of the first three months of 2010?

    -The approximate amount of solid waste in the landfill at the end of the first three months of 2010 is 1,411 tons, obtained by using the slope of the tangent line at t equals 0 and projecting forward by 1/4 of a year.

  • Why is the approximation at the end of the first three months not the exact amount?

    -The approximation is not the exact amount because it relies on the initial slope and assumes a linear increase without considering the potential non-linearities of the actual function W over time.

  • How does the differential equation provided help in understanding the growth of solid waste in the landfill?

    -The differential equation provides a model for the rate of change of the solid waste in the landfill over time, allowing for predictions and approximations of future waste amounts based on the initial conditions and the given growth rate.

  • What is the role of the parameter '300' in the differential equation?

    -The parameter '300' in the differential equation represents a threshold value that the function W must reach before any additional waste is added to the landfill, according to the model.

  • What is the unit of time used in the differential equation?

    -The unit of time used in the differential equation is years, specifically measuring from the start of the year 2010.

Outlines
00:00
📈 Initial Waste Analysis and Tangent Line Application

This paragraph introduces the problem of estimating the amount of solid waste in a landfill at the end of the first three months of 2010, using a given differential equation. The landfill initially contains 1,400 tons of waste, and the total amount of waste, represented by the function W, is expected to increase over the next 20 years according to the equation dW/dt = 1/25 * (W - 300). The task is to use the tangent line to the graph of W at t=0 to approximate the waste amount after a quarter of a year. The explanation involves understanding the increasing nature of the function W, calculating the slope of the tangent line at t=0 using the derivative from the differential equation, and then using this slope to estimate the waste quantity after 1/4 year. The slope is determined to be 44, leading to an estimated waste amount of 1,411 tons after three months.

05:01
📊 Utilizing the Tangent Line Slope for Waste Estimation

The second paragraph focuses on applying the calculated slope of the tangent line to estimate the waste tonnage after 1/4 of a year. It reiterates the slope of 44 from the previous calculation and explains how this slope represents the increase in waste tonnage per unit time. The paragraph then uses this information to approximate the waste amount after a quarter of a year by considering the linear nature of the tangent line approximation. The equation of the tangent line is formulated as W_approx = 44 * t + 1,400, and by substituting t with 1/4, the estimated waste amount is calculated to be 1,411 tons. This approximation, while not exact, provides a reasonable estimate based on the given data and the tangent line method.

Mindmap
Keywords
💡Landfill
A landfill is a designated area for the disposal of waste materials. In the context of the video, it is the location where solid waste is stored and managed. The script mentions a landfill that initially contained 1,400 tons of solid waste at the beginning of 2010, highlighting the environmental and waste management issues addressed in the video.
💡Increasing Function
An increasing function is a mathematical function that grows larger as its independent variable (often time) increases. In the video, the function W models the total amount of solid waste in the landfill, and it is described as an increasing function, indicating that the amount of waste is expected to rise over the 20-year period.
💡Differential Equation
A differential equation is an equation that relates a function to its derivatives. In the context of the video, the differential equation describes the rate of change of the waste accumulation function W with respect to time. The equation is used to model the increasing amount of waste in the landfill.
💡Derivative
The derivative of a function is a measure of how the function changes as its input changes. It is a fundamental concept in calculus and is used in the video to determine the rate at which the waste amount in the landfill is increasing or decreasing at any given time.
💡Tangent Line
A tangent line is a line that touches a curve at a single point and has the same slope as the curve at that point. In the video, the tangent line to the graph of W at t equals 0 is used to approximate the amount of waste at the end of the first three months of 2010, providing a linear approximation of the function's behavior.
💡Solid Waste
Solid waste refers to any garbage or refuse that is not liquid. In the video, solid waste is the focus of the problem, as it discusses the accumulation of such waste in a landfill over time, which is a critical issue in environmental management.
💡Slope
The slope of a line is a measure of its steepness, indicating the rate of change between the vertical and horizontal distances. In the context of the video, the slope of the tangent line at t equals 0 is used to approximate the future value of the waste accumulation function W.
💡Approximation
An approximation is a rough estimate or a simplified version of a value or function. In the video, the approximation is used to estimate the amount of solid waste in the landfill after three months, based on the initial conditions and the function's behavior.
💡Time
Time, in this context, is the independent variable used to measure the accumulation of waste. It is a fundamental aspect of the differential equation that models the waste accumulation function W, with the video considering a 20-year period.
💡Waste Management
Waste management refers to the collection, transportation, disposal, or recycling of waste, in an effort to reduce its effect on human health and the environment. The video's focus on the accumulation of solid waste in a landfill and the modeling of this accumulation over time is directly related to waste management practices and their environmental impact.
💡Environmental Issues
Environmental issues are concerns related to the natural environment's well-being and the impact of human activities upon it. The video addresses an environmental issue by focusing on the accumulation of solid waste in a landfill, which is a significant problem affecting environmental health.
Highlights

The landfill contained 1,400 tons of solid waste at the beginning of 2010.

The total amount of solid waste stored at the landfill is modeled by an increasing function W.

The derivative of W with respect to time is given by a differential equation involving W(t) - 300.

The time frame for the waste accumulation is over the next 20 years.

W is measured in tons and t in years from the start of 2010.

Part a of the problem asks to approximate the amount of waste at the end of the first 3 months (1/4 year) of 2010 using the tangent line to W at t=0.

The landfill's waste accumulation function W is increasing and starts at 1,400 tons.

The slope of the tangent line at t=0 is the derivative of W evaluated at that point.

The derivative of W at t=0 is calculated as 1/25 times the initial waste (1,400 tons) minus 300.

The slope (m) of the tangent line at t=0 is found to be 44.

The tangent line approximation is used to estimate the waste amount after 1/4 year.

The equation of the tangent line approximation is W = 44t + 1400.

The estimated waste amount at the end of the first 3 months is 1,411 tons.

The approximation is based on the initial condition and the calculated slope of the tangent line.

The approximation may not be exact but serves as a reasonable estimate.

Transcripts

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Mathematical ModelingDifferential EquationsWaste ManagementLandfill GrowthEnvironmental ScienceData AnalysisApproximation TechniquesTime SeriesSustainabilityResource Management