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

$A = P e^{rt}$

Where:

• $A$ is the final amount.
• $P$ is the initial principal (starting amount).
• $r$ is the annual interest rate (in decimal form).
• $t$ is the time in years.
• $e$ is Euler's number (approximately 2.71828).

Since the investment will triple, we know that $A = 3P$. Plugging this into the formula:

$3P = P e^{0.07t}$

We can divide both sides by $P$ (as long as $P \neq 0$):

$3 = e^{0.07t}$

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

$\ln(3) = 0.07t$

Now, solve for $t$:

$t = \frac{\ln(3)}{0.07}$

Using $\ln(3) \approx 1.0986$:

$t = \frac{1.0986}{0.07} \approx 15.7 \, \text{years}$

So, it will take approximately 15.7 years for the investment to triple.

Would you like further details on any part of the solution? Here are some related questions to explore:

1. What if the interest rate were 5% instead of 7%?
2. How long would it take for the investment to double at 7% interest?
3. What is the formula for compounded interest if it is not continuous?
4. What is Euler's number, and how is it derived?
5. How does compounding frequency (monthly, quarterly, etc.) affect growth?

Tip: Continuous compounding grows the investment faster than regular compounding due to the constant application of interest.