Modeling the World: Exponential Growth and Decay

Algebra 2 Grades High School 10:22 Video
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Lesson Description

Explore exponential growth and decay models using real-world examples. Learn to apply the formulas y = ae^(bx) and y = ae^(-bx) to solve problems involving bacterial growth and radioactive decay.

Video Resource

Exponential Growth and Decay Models y=ae^bx and y=ae^-bx

Mario's Math Tutoring

Duration: 10:22
Watch on YouTube

Key Concepts

  • Exponential Growth
  • Exponential Decay
  • Natural Logarithms
  • Mathematical Modeling

Learning Objectives

  • Students will be able to identify and apply the appropriate exponential growth or decay model to a given scenario.
  • Students will be able to solve for unknown variables in exponential growth and decay models using logarithms.
  • Students will be able to interpret the results of exponential growth and decay calculations in the context of real-world problems.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general forms of exponential growth and decay models: y = ae^(bx) and y = ae^(-bx). Discuss the meaning of each variable (a, b, x, y) and their significance in the context of growth and decay.
  • Video Presentation (15 mins)
    Play the video "Exponential Growth and Decay Models y=ae^bx and y=ae^-bx" by Mario's Math Tutoring. Instruct students to take notes on the examples provided, focusing on how the initial conditions are used to determine the parameters of the model.
  • Worked Examples and Practice (20 mins)
    Work through additional examples, similar to those in the video, as a class. Emphasize the steps involved in setting up the equation, solving for the unknown rate (b), and using the model to make predictions. Provide students with practice problems to solve independently or in small groups.
  • Discussion and Wrap-up (10 mins)
    Facilitate a class discussion to address any remaining questions or misconceptions. Summarize the key concepts covered in the lesson and highlight the applications of exponential growth and decay in various fields.

Interactive Exercises

  • Bacterial Growth Simulation
    Use an online simulation or spreadsheet software to model bacterial growth under different conditions. Students can manipulate the initial population and growth rate to observe the effects on the population size over time.
  • Radioactive Decay Activity
    Provide students with data on the half-life of a radioactive substance. Have them use this information to create an exponential decay model and predict the amount of substance remaining after a given period.

Discussion Questions

  • How does the sign of 'b' in the exponential model determine whether it's growth or decay?
  • What are some other real-world scenarios that can be modeled using exponential growth and decay?

Skills Developed

  • Mathematical Modeling
  • Problem Solving
  • Logarithmic Manipulation
  • Data Interpretation

Multiple Choice Questions

Question 1:

Which of the following equations represents exponential decay?

Correct Answer: y = ae^(bx), where b < 0

Question 2:

In the exponential growth model y = ae^(bx), 'a' represents the:

Correct Answer: Initial amount

Question 3:

What is the inverse operation used to solve for an exponent in an exponential equation?

Correct Answer: Logarithm

Question 4:

A population of bacteria doubles every 3 hours. What type of model best represents this scenario?

Correct Answer: Exponential Growth

Question 5:

A radioactive substance decays at a rate proportional to its current amount. This is an example of:

Correct Answer: Exponential decay

Question 6:

If the half-life of a substance is 10 years, how long will it take for the amount of the substance to reduce to 25% of its initial amount?

Correct Answer: 20 years

Question 7:

What is the purpose of using the natural logarithm (ln) in solving exponential growth and decay problems?

Correct Answer: To isolate the exponent when the base is 'e'

Question 8:

The formula for continuous compound interest, A=Pe^(rt), is an example of?

Correct Answer: Exponential growth

Question 9:

What does the variable 'x' generally represent in the exponential growth and decay models discussed?

Correct Answer: Time

Question 10:

If you take the natural log of both sides of e^(2x) = 5, what is the next step to solve for 'x'?

Correct Answer: Divide both sides by 2

Fill in the Blank Questions

Question 1:

The formula y = ae^(bx) represents exponential ________ if b > 0.

Correct Answer: growth

Question 2:

In the formula y = ae^(-bx), the negative sign indicates exponential ________.

Correct Answer: decay

Question 3:

The initial amount in an exponential growth or decay problem is represented by the variable ________.

Correct Answer: a

Question 4:

The constant 'e' is known as the ________ base.

Correct Answer: natural

Question 5:

To solve for an exponent when the base is 'e', you use the ________ logarithm.

Correct Answer: natural

Question 6:

The time it takes for a radioactive substance to decay to half of its original amount is called the ________.

Correct Answer: half-life

Question 7:

In the formula A = Pe^(rt) for continuous compound interest, 'r' represents the ________ rate.

Correct Answer: interest

Question 8:

If a population's growth can be modeled by an exponential function, and it starts at 500 and doubles every hour, the amount after 't' hours would be expressed as y=500*2^________.

Correct Answer: t

Question 9:

The general solution to finding the value of b in these problems requires the use of a ________.

Correct Answer: logarithm

Question 10:

Log base e is also known as ________.

Correct Answer: ln

Educational Standards

A2.A.1.2:Use mathematical models to represent exponential relationships, such as compound interest, depreciation, and population growth. Solve these equations algebraically or graphically (including graphing calculator or other appropriate technology). A2.F.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms.

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