Tripling Your Investment: Mastering Continuous Compounding

Algebra 2 Grades High School 2:58 Video
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Lesson Description

This lesson explores the concept of continuous compounding and how to calculate the time it takes for an investment to triple. Students will learn to apply the continuous compounding formula and use natural logarithms to solve for time.

Video Resource

Time to Triple Compounding Continuously

Mario's Math Tutoring

Duration: 2:58
Watch on YouTube

Key Concepts

  • Continuous Compounding Formula: A = Pe^(rt)
  • Natural Logarithms (ln)
  • Solving Exponential Equations Using Logarithms

Learning Objectives

  • Students will be able to apply the continuous compounding formula to calculate the future value of an investment.
  • Students will be able to use natural logarithms to solve for the time it takes for an investment to triple at a given interest rate.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of compound interest and contrasting it with continuous compounding. Introduce the continuous compounding formula: A = Pe^(rt), where A is the future value, P is the principal, r is the interest rate, and t is time. Briefly discuss the number 'e' and its significance.
  • Video Viewing (5 mins)
    Watch the video 'Time to Triple Compounding Continuously' by Mario's Math Tutoring (https://www.youtube.com/watch?v=VFkHNsjyZys). Pay attention to the example problem and the steps taken to solve it.
  • Example Problem Walkthrough (10 mins)
    Work through the example problem from the video on the board, emphasizing each step: setting up the equation, isolating the exponential term, applying the natural logarithm, and solving for t. Discuss why the natural logarithm is used.
  • Practice Problems (15 mins)
    Provide students with practice problems where they calculate the time it takes for different investments to triple at various interest rates. Circulate to provide assistance and answer questions.
  • Review and Wrap-up (5 mins)
    Review the key concepts and the steps involved in solving for time in continuous compounding problems. Address any remaining questions and preview the upcoming quiz.

Interactive Exercises

  • Rate Comparison
    Present scenarios with different interest rates. Students calculate and compare the time it takes for an investment to triple at each rate, discussing the impact of interest rate on growth.

Discussion Questions

  • How does continuous compounding differ from compounding monthly or annually?
  • Why is the natural logarithm used when solving for time in the continuous compounding formula?
  • How does the principal amount affect the time it takes to triple your investment with continuous compounding?
  • How would the formula change if you were trying to determine the time to double the initial investment instead of triple?

Skills Developed

  • Applying the continuous compounding formula
  • Solving exponential equations using natural logarithms
  • Problem-solving skills

Multiple Choice Questions

Question 1:

The continuous compounding formula is given by:

Correct Answer: A = Pe^(rt)

Question 2:

What does 'e' represent in the continuous compounding formula?

Correct Answer: Euler's number (approximately 2.71828)

Question 3:

If an investment triples, the future value (A) is equal to:

Correct Answer: 3P

Question 4:

To solve for 't' in the continuous compounding formula, you need to use:

Correct Answer: Natural logarithm

Question 5:

What is the purpose of using natural logs in solving for time?

Correct Answer: To undo the exponential function with base e

Question 6:

Which of the following is closest to the value of ln(3)?

Correct Answer: 1.1

Question 7:

What effect does a higher interest rate have on the time it takes to triple an investment?

Correct Answer: Decreases the time

Question 8:

What value should replace 'A' if you want to find the time it takes to double the initial investment?

Correct Answer: 2P

Question 9:

What is the first step in finding the time it takes to triple an investment using the continuous compounding formula?

Correct Answer: Substitute the known values into the formula

Question 10:

Which variable in the continuous compounding formula does not affect the time it takes to triple the investment?

Correct Answer: P

Fill in the Blank Questions

Question 1:

The formula for continuous compounding is A = P * e^(____)

Correct Answer: rt

Question 2:

If an investment triples, the future value (A) is ____ times the principal (P).

Correct Answer: 3

Question 3:

The inverse function of e^x is ____.

Correct Answer: ln(x)

Question 4:

To solve for time (t) when the investment triples, you need to divide both sides of the equation by the natural log of ____ .

Correct Answer: e

Question 5:

The approximate value of 'e' is ____.

Correct Answer: 2.71828

Question 6:

ln(e) equals ____.

Correct Answer: 1

Question 7:

A higher interest rate results in a ____ amount of time to triple an investment.

Correct Answer: shorter

Question 8:

The natural log is a log with a base of ____.

Correct Answer: e

Question 9:

The first step to finding the time to triple your investment is to substitute all known values including ____ for 'A'.

Correct Answer: 3P

Question 10:

The continuous compounding formula is best applied when interest is compounded ____.

Correct Answer: continuously

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.

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