Problem 5
Question
If a bank pays \(6 \%\) per year interest compounded continuously, how long does it take for the balance in an account to double?
Step-by-Step Solution
Verified Answer
It takes approximately 11.55 years for the balance to double.
1Step 1: Understanding Continuous Compounding
In continuous compounding, the formula for the amount after time \( t \) is given by \( A = Pe^{rt} \), where \( A \) is the amount, \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years. We want to find how long, \( t \), it takes for the principal, \( P \), to double.
2Step 2: Setting Up the Equation
Since we want the balance to double, let \( A = 2P \). Substituting into the formula, we have \( 2P = Pe^{0.06t} \). The \( P \) will cancel out from both sides, simplifying to \( 2 = e^{0.06t} \).
3Step 3: Solving for t Using Natural Logs
To solve for \( t \), take the natural logarithm of both sides: \( \ln(2) = \ln(e^{0.06t}) \). This simplifies to \( \ln(2) = 0.06t \), as the logarithm of an exponential function brings down the exponent.
4Step 4: Calculating t
Now solve for \( t \): \( t = \frac{\ln(2)}{0.06} \). Calculate the natural logarithm, \( \ln(2) \approx 0.6931 \). Thus, \( t \approx \frac{0.6931}{0.06} \approx 11.55 \).
Key Concepts
Interest RateNatural LogarithmsExponential Growth
Interest Rate
An interest rate is essentially what you earn or pay for the use of money over time. There are different ways that interest can be compounded, which determines how much you ultimately earn or owe. In continuous compounding, interest is calculated and added to the balance at every moment in time, which makes it different from traditional compounding periods like annually, monthly, or daily.
This means that the balance grows faster because interest is constantly being applied to the running total.
When solving problems involving continuous compounding, you'll come across an interest rate written as a decimal in the formula: \[ A = Pe^{rt} \] where:
This means that the balance grows faster because interest is constantly being applied to the running total.
When solving problems involving continuous compounding, you'll come across an interest rate written as a decimal in the formula: \[ A = Pe^{rt} \] where:
- \( A \) is the future amount of money you'll have
- \( P \) represents the initial principal balance
- \( e \) is the base of the natural logarithm, approximately equal to 2.718
- \( r \) is the annual interest rate
- \( t \) is the time in years
Natural Logarithms
Natural logarithms are a type of logarithm that uses the base \( e \), which is approximately 2.718. They are denoted as \( \ln \). Natural logarithms are particularly useful when dealing with exponential growth, like in our continuous compounding example.
A logarithm tells you how many times you need to multiply the base to get a certain number.
In mathematical problems, ensuring the equality of expressions often requires using logarithms. For our problem:
A logarithm tells you how many times you need to multiply the base to get a certain number.
In mathematical problems, ensuring the equality of expressions often requires using logarithms. For our problem:
- If you have the equation \( 2 = e^{0.06t} \), you can apply the logarithm to both sides to solve for \( t \).
- It turns \( 2 = e^{0.06t} \) into \( \ln(2) = 0.06t \).
Exponential Growth
Exponential growth is a process where an amount increases by a consistent percentage over regular intervals. Continuous compounding is a prime example of exponential growth.
In this context, while traditional compound interest might be calculated annually or monthly, continuous compounding implies that growth never stops—it's always happening, no matter how small the time interval.
The formula \( A = Pe^{rt} \) is a classic representation of exponential growth. Here the principal amount grows continuously at the given interest rate \( r \) over time \( t \).
Understanding exponential growth is crucial when analyzing investments, as it affects how quickly an investment can grow.
For example, due to continuously compounding interest, even small differences in interest rates can lead to significant differences in future balances over time.
In this context, while traditional compound interest might be calculated annually or monthly, continuous compounding implies that growth never stops—it's always happening, no matter how small the time interval.
The formula \( A = Pe^{rt} \) is a classic representation of exponential growth. Here the principal amount grows continuously at the given interest rate \( r \) over time \( t \).
Understanding exponential growth is crucial when analyzing investments, as it affects how quickly an investment can grow.
For example, due to continuously compounding interest, even small differences in interest rates can lead to significant differences in future balances over time.
- Continuously compounding utilizes the exponential function \( e^{rt} \) which grows faster than simple or regular compound interest over the same period.
- Thus, with exponential growth, your investment could potentially double in a noticeably shorter time frame.
Other exercises in this chapter
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