Question: How long does it take for a $25,000 investment take to triple when invested at 6.5% interest compounded continuously?
1. **State the problem:** We want to find the time $t$ it takes for an investment of $25,000$ to triple at an interest rate of $6.5\%$ compounded continuously.
2. **Formula used:** The formula for continuous compounding is $$A = P e^{rt}$$ where:
- $A$ is the amount after time $t$,
- $P$ is the principal (initial amount),
- $r$ is the annual interest rate (as a decimal),
- $t$ is the time in years,
- $e$ is Euler's number (approximately $2.71828$).
3. **Set up the equation:** Since the investment triples, $A = 3P$. Substitute into the formula:
$$3P = P e^{rt}$$
4. **Simplify:** Divide both sides by $P$:
$$3 = e^{rt}$$
5. **Take the natural logarithm of both sides:**
$$\ln(3) = rt$$
6. **Solve for $t$:**
$$t = \frac{\ln(3)}{r}$$
7. **Substitute $r = 6.5\% = 0.065$:**
$$t = \frac{\ln(3)}{0.065}$$
8. **Calculate:**
$$\ln(3) \approx 1.0986$$
$$t = \frac{1.0986}{0.065} \approx 16.9$$
**Answer:** It takes approximately **16.9 years** for the investment to triple with continuous compounding at $6.5\%$ interest.