To determine how long it will take for an initial investment of $\$500$ to grow to $\$1000$ at an interest rate of $4\%$ compounded continuously, we can use the formula for continuous compounding:

$A = P e^{rt}$

where:

• $A$ is the amount of money accumulated after time $t$,
• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate,
• $t$ is the time the money is invested for,
• $e$ is the base of the natural logarithm.

Given:

• $A = 1000$
• $P = 500$
• $r = 0.04$

We need to find $t$.

1. Substitute the given values into the continuous compounding formula:

$1000 = 500 e^{0.04t}$

2. Divide both sides by 500 to isolate the exponential term:

$2 = e^{0.04t}$

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

$\ln(2) = \ln(e^{0.04t})$

4. Use the property of logarithms $\ln(e^x) = x$:

$\ln(2) = 0.04t$

5. Solve for $t$:

$t = \frac{\ln(2)}{0.04}$

6. Calculate $\ln(2)$ using a calculator:

$\ln(2) \approx 0.693$

7. Substitute back into the equation:

$t = \frac{0.693}{0.04}$

8. Perform the division:

$t \approx 17.325$

Therefore, it will take approximately 17.325 years for the investment to grow to $\$1000$ with continuous compounding at an interest rate of $4\%$.

Would you like further details or have any questions about this calculation?

Here are five related questions to consider:

1. How would the time change if the interest rate were increased to $5\%$?
2. What would the final amount be if the initial investment were $\$1000$ at the same interest rate for 17.325 years?
3. How does continuous compounding compare to annual compounding in terms of total amount after 17.325 years?
4. What is the formula for calculating the time to double an investment with continuous compounding?
5. How would you calculate the amount in the account after 10 years with continuous compounding at $4\%$?

Tip: When solving exponential growth problems, taking the natural logarithm of both sides is a powerful technique to isolate the variable of interest.