Unlocking Half-Life: Mastering Exponential Decay

Algebra 2 Grades High School 2:45 Video
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Lesson Description

Explore the concept of half-life, its formula, and practical applications. Learn to solve half-life problems using logarithms and the change of base formula.

Video Resource

Half Life Formula & Example

Mario's Math Tutoring

Duration: 2:45
Watch on YouTube

Key Concepts

  • Half-Life Formula (N = N₀ * (1/2)^(t/T))
  • Exponential Decay
  • Logarithms and their properties
  • Change of Base Formula

Learning Objectives

  • Students will be able to apply the half-life formula to solve for remaining amount, time, initial amount, or half-life.
  • Students will be able to use logarithms to solve for the time variable in half-life problems.
  • Students will be able to apply the change of base formula to evaluate logarithms with bases other than 10 or e.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of exponential decay. Ask students for examples of real-world phenomena that exhibit exponential decay. Introduce the term 'half-life' as the time it takes for a quantity to reduce to half of its initial value.
  • Formula Explanation (5 mins)
    Present the half-life formula: N = N₀ * (1/2)^(t/T), where N is the final amount, N₀ is the initial amount, t is the time elapsed, and T is the half-life. Clearly define each variable and its significance. Emphasize that (1/2) represents the decay factor.
  • Example Problem (15 mins)
    Work through the example problem from the video: 'You have 100g of a substance with a half-life of 20 days. How long will it take to have 10g remaining?' Guide students step-by-step through setting up the equation, isolating the exponential term, taking the logarithm of both sides, and using the change of base formula. Stress the importance of showing each step clearly.
  • Practice Problems (15 mins)
    Provide students with additional practice problems of varying difficulty. Encourage them to work in pairs or small groups. Circulate to provide assistance and answer questions. Example problems: 1. A radioactive isotope has a half-life of 5 years. How much of a 200g sample will remain after 15 years? 2. A substance decays from 50g to 12.5g in 8 hours. What is its half-life?
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts covered in the lesson. Answer any remaining questions from the students. Assign homework that reinforces the concepts and provides further practice.

Interactive Exercises

  • Half-Life Simulation
    Use an online simulation (e.g., from PhET Interactive Simulations) to visualize exponential decay and the concept of half-life. Allow students to manipulate variables and observe the effects on the decay process.
  • Group Problem Solving
    Divide the class into groups and assign each group a complex half-life problem to solve. Have each group present their solution to the class, explaining their reasoning and calculations.

Discussion Questions

  • Can you think of other real-world examples where the concept of half-life might be applicable?
  • Why is it necessary to use logarithms to solve for the time variable in half-life problems?
  • How does the change of base formula allow us to use calculators that only have common or natural logarithm functions?

Skills Developed

  • Problem-solving
  • Algebraic manipulation
  • Logarithmic functions
  • Critical thinking

Multiple Choice Questions

Question 1:

What does 'N₀' represent in the half-life formula N = N₀ * (1/2)^(t/T)?

Correct Answer: The initial amount

Question 2:

In the half-life formula, what is the base of the exponential term?

Correct Answer: 1/2

Question 3:

What mathematical function is used to solve for 't' when it is in the exponent in the half-life formula?

Correct Answer: Logarithm

Question 4:

What is the purpose of the change of base formula?

Correct Answer: To convert between different logarithmic bases

Question 5:

A substance has a half-life of 10 years. If you start with 50 grams, how much will remain after 20 years?

Correct Answer: 12.5 grams

Question 6:

Which of the following is an example of exponential decay?

Correct Answer: Radioactive decay

Question 7:

What is the value of log₂(8)?

Correct Answer: 4

Question 8:

If the half-life of a substance is 5 days, how many half-lives occur in 15 days?

Correct Answer: 3

Question 9:

The half-life of a radioactive element is 30 minutes. What fraction of the original sample remains after 1 hour?

Correct Answer: 1/4

Question 10:

Which of these equations represents exponential decay?

Correct Answer: y = (1/2)^x

Fill in the Blank Questions

Question 1:

The time it takes for a substance to reduce to half of its initial amount is called its ________.

Correct Answer: half-life

Question 2:

The half-life formula is N = ________ * (1/2)^(t/T).

Correct Answer: N₀

Question 3:

If the variable you are solving for is in the exponent, you need to use ________ to solve the equation.

Correct Answer: logarithms

Question 4:

The change of base formula allows you to calculate logarithms with bases other than 10 or ________.

Correct Answer: e

Question 5:

A substance decaying over time is an example of ________ decay.

Correct Answer: exponential

Question 6:

If a substance has a half-life of 4 years, then after 8 years, ________ half-lives have occurred.

Correct Answer: 2

Question 7:

In the half-life formula, 't' represents the ________ ________.

Correct Answer: time elapsed

Question 8:

When taking the logarithm of both sides of an equation, it is important to keep the equation ________.

Correct Answer: balanced

Question 9:

The inverse operation of exponentiating is taking the ________.

Correct Answer: logarithm

Question 10:

The number 1/2 in the half-life formula represents the ________ ________.

Correct Answer: decay factor

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.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms. 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.

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