Problem 35 - Intermediate Algebra

Question

How long will it take $$\$ 2000$$ to double itself if it is invested at \(6 \%\) interest compounded continuously?

Step-by-Step Solution

Answer

About 11.55 years.

Step 1: Understanding the Problem

We are asked to find out how long it will take for $2000 to double at an interest rate of 6% compounded continuously.

Step 2: Use the Formula for Continuous Compounding

When interest is compounded continuously, the formula to use is \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount, \( r \) is the interest rate, and \( e \) is the base of the natural logarithm. In this problem, we want \( A = 2P \).

Step 3: Set Up the Equation

To find the time \( t \), we set the equation as \( 2P = Pe^{0.06t} \). Here, \( r = 0.06 \) and \( A = 2P \).

Step 4: Simplify and Solve for \( t \)

Divide both sides by \( P \) to get \( 2 = e^{0.06t} \). To solve for \( t \), take the natural logarithm of both sides: \( \ln 2 = 0.06t \).

Step 5: Calculate \( t \)

Solve for \( t \) by dividing both sides by \( 0.06 \): \( t = \frac{\ln 2}{0.06} \). Using a calculator, \( \ln 2 \approx 0.693 \), so \( t \approx \frac{0.693}{0.06} \).

Step 6: Final Calculation

Now calculate \( t \approx 11.55 \). This means it will take approximately 11.55 years for the investment to double.

Key Concepts

Continuous Compounding

Continuous compounding is a powerful financial concept. It refers to the scenario where an unlimited number of compounding periods are applied within a certain time frame. Unlike regular compounding, which might be monthly or annually, continuous compounding assumes the interest is always growing, with time broken down into infinitely small intervals.

In mathematical terms, the formula used specific to continuous compounding is:

• \[ A = Pe^{rt} \]

where:

• \( A \) is the amount of money accumulated after time \( t \)
• \( P \) is the principal or initial amount
• \( r \) is the annual interest rate (as a decimal)
• \( t \) is the time in years
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

Continuous compounding allows your money to grow faster than with standard compounding methods, making it highly beneficial in long-term investments. Understanding this concept is crucial when evaluating investment opportunities that utilize continuous compounding.

Exponential Growth

Exponential growth describes a process that increases at a consistent rate over time. In the context of finance and investments, it reflects how an investment can grow increasingly faster over time due to the effects of compounding.

With exponential growth, as seen in continuous compounding scenarios, growth accelerates as it builds upon the previously accrued amount, rather than just the initial principal.

To grasp exponential growth in investments, consider savings where interest continually reinvests. Each reinvestment becomes a new foundation for further growth, leading to a curve that becomes steeper over time. This differs from linear growth, where increases remain constant.

Understanding exponential growth helps investors recognize the impact of interest rates over long periods and why starting early can significantly multiply the benefits of an investment.

Natural Logarithms

Natural logarithms are fundamental in solving exponential equations, making them essential for continuous compounding calculations. A natural logarithm is the logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828.

In continuous compounding, the natural logarithm is used to solve for time when evaluating how long it will take an investment to reach a certain value. For example, when doubling an investment using continuous compounding, we use the natural logarithm as follows:

• To isolate \( t \) in \( 2 = e^{0.06t} \), take the natural logarithm of both sides: \( \ln 2 = 0.06t \).
• Further solving gives \( t = \frac{\ln 2}{0.06} \), showing the simplicity and elegance of logarithms in breaking down complex exponential problems.

Natural logarithms simplify solving exponential equations, turning multiplication or powers into manageable addition, providing a clear pathway to understanding growth rates and investment timelines.