https://file.aiquickdraw.com/mathbot/file/1728830616456pldv7vo3.jpg

To solve this problem, we'll use the formula for continuous compounding, which is:

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

where:
- \( A \) = the final amount after time \( t \),
- \( P \) = the initial principal (the original amount),
- \( r \) = the interest rate (as a decimal),
- \( t \) = time in years,
- \( e \) = Euler's number (approximately 2.71828).

Given that the investment triples, we can set \( A = 3P \), \( r = 0.05 \) (5% interest), and solve for \( t \). This results in the equation:

\[
3P = P e^{0.05t}
\]

Dividing both sides by \( P \), we get:

\[
3 = e^{0.05t}
\]

Taking the natural logarithm (ln) on both sides, we have:

\[
\ln(3) = 0.05t
\]

Now, solve for \( t \):

\[
t = \frac{\ln(3)}{0.05}
\]

Let's calculate this:

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

\[
t = \frac{1.0986}{0.05} \approx 21.97
\]

So, the time \( t \) is approximately **21.97 years**.

Would you like a more detailed explanation or have any other questions?

Here are some related questions to consider:
1. What is the formula for continuous compounding?
2. How does changing the interest rate affect the time required for an investment to triple?
3. What is the difference between continuous compounding and simple interest?
4. How do you find the natural logarithm of a number?
5. What would happen if the investment only doubled instead of tripled?

**Tip:** When dealing with continuous compounding, the natural logarithm (ln) is a crucial tool to solve equations involving \( e \).

An investment in a savings account grows to three times the initial value after t years. If the rate of interest is 5%, compounded continuously, what is t?

Learn how to calculate the time it takes for an investment to triple with a 5% interest rate compounded continuously. This problem uses the continuous compound interest formula A = P e^(rt) and involves natural logarithms to find the solution.

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