How long will it take for an investment to​ triple, if interest is compounded continuously at 8​%?
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Part 1
It will take

enter your response here years before the investment triples.

To determine how long it will take for an investment to triple with continuous compounding interest, we use the formula for continuous compounding:

\[ A = P e^{rt} \]

where:
- \( A \) is the final amount
- \( P \) is the initial principal
- \( r \) is the annual interest rate (expressed as a decimal)
- \( t \) is the time in years
- \( e \) is the base of the natural logarithm

In this problem, we want to find the time \( t \) when the investment triples, so \( A = 3P \). The interest rate is 8%, or \( r = 0.08 \). The equation becomes:

\[ 3P = P e^{0.08t} \]

We can simplify this by dividing both sides by \( P \):

\[ 3 = e^{0.08t} \]

To solve for \( t \), take the natural logarithm of both sides:

\[ \ln(3) = 0.08t \]

Now, solve for \( t \):

\[ t = \frac{\ln(3)}{0.08} \]

Using the natural logarithm of 3:

\[ \ln(3) \approx 1.0986 \]

So,

\[ t = \frac{1.0986}{0.08} \approx 13.7325 \]

Therefore, it will take approximately 13.73 years for the investment to triple if the interest is compounded continuously at 8%.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the formula for calculating compound interest for different compounding periods (e.g., annually, quarterly)?
2. How does the continuous compounding formula differ from other compounding formulas?
3. What is the significance of the natural logarithm in the context of continuous compounding?
4. How does changing the interest rate affect the time it takes for an investment to triple?
5. What is the relationship between the interest rate and the time needed to reach a certain amount with continuous compounding?
6. Can you derive the continuous compounding formula from basic principles?
7. How does the time to triple change if the investment is compounded annually instead of continuously?
8. What are some real-world applications of continuous compounding?

**Tip:** Understanding the properties of exponential functions and natural logarithms is crucial when dealing with continuously compounded interest problems.

Learn how to calculate the time it takes for an investment to triple with continuous compounding at an annual interest rate of 8%. Understand the formula A = P e^(rt) and solve for the time using natural logarithms. Suitable for advanced high school students or introductory college-level students.

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