Adam invests $$ \$ 6,139 $$ in a retirement account with a fixed annual interest rate compounded continuously. After 17 years, the balance reaches $$ \$ 8,624.97 $$. What is the interest rate of the account?

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To find the interest rate of the account, we can use the formula for continuous compounding, which is given by:

$$
A = Pe^{rt}
$$

where:
- $$ A $$ is the amount of money accumulated after time $$ t $$,
- $$ P $$ is the principal amount (the initial amount of money),
- $$ r $$ is the annual interest rate (in decimal),
- $$ t $$ is the time the money is invested for (in years),
- $$ e $$ is the base of the natural logarithm (approximately equal to 2.71828).

In this case, we have:
- $$ A = 8624.97 $$
- $$ P = 6139 $$
- $$ t = 17 $$

We need to solve for $$ r $$. Plugging in the values we have:

$$
8624.97 = 6139 e^{17r}
$$

To isolate $$ e^{17r} $$, we divide both sides by 6139:

$$
e^{17r} = \frac{8624.97}{6139}
$$

Calculating the right side:

$$
e^{17r} \approx 1.404
$$

Next, we take the natural logarithm of both sides:

$$
\ln(e^{17r}) = \ln(1.404)
$$

Using the property of logarithms that $$ \ln(e^x) = x $$:

$$
17r = \ln(1.404)
$$

Now we calculate $$ \ln(1.404) $$:

$$
\ln(1.404) \approx 0.339
$$

Now we can solve for $$ r $$:

$$
17r = 0.339
$$

$$
r = \frac{0.339}{17} \approx 0.01994
$$

To express $$ r $$ as a percentage, we multiply by 100:

$$
r \approx 0.01994 \times 100 \approx 1.994\%
$$

Thus, the interest rate of the account is approximately:

$$
\boxed{1.99\%}
$$
The interest rate of the account is approximately 1.99%.

Adam invests in a retirement
account with a fixed annual interest rate
compounded continuously. After 17 years,
the balance reaches . What is the
interest rate of the account?

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Adam invests in a retirement account with a fixed annual interest rate compounded continuously. After 17 years, the balance reaches . What is the interest rate of the account?