7400 dollars is placed in an account with an annual interest rate of 6.25%. To the nearest tenth of a year, how long will it take for the account value to reach 39200 dollars?

We can solve this problem using the formula for compound interest, specifically the continuous compound interest formula:

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

Where:
- $ A $ is the final amount (39200 dollars),
- $ P $ is the initial principal (7400 dollars),
- $ r $ is the annual interest rate (6.25% or 0.0625),
- $ t $ is the time in years, and
- $ e $ is Euler's number (approximately 2.71828).

We need to solve for $ t $. First, divide both sides of the equation by $ P $:

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

Now take the natural logarithm (ln) of both sides to get rid of the exponent:

\[
\ln\left(\frac{A}{P}\right) = rt
\]

Finally, solve for $ t $:

\[
t = \frac{\ln\left(\frac{A}{P}\right)}{r}
\]

Now, substitute the given values:

\[
t = \frac{\ln\left(\frac{39200}{7400}\right)}{0.0625}
\]

Let me calculate this.The time it will take for the account value to reach 39,200 dollars is approximately 26.7 years.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the formula for compound interest if compounded monthly instead of continuously?
2. How much interest will be earned after 10 years at the same rate and principal?
3. If the interest rate were 7%, how would the time to reach 39,200 dollars change?
4. What is the value of the account after 15 years with a 6.25% annual interest rate?
5. How does continuous compounding compare with annual compounding?

**Tip**: Continuous compounding grows the investment faster than annual or monthly compounding due to the effect of constantly earning interest on the accumulated interest.

Learn how to calculate the time needed for $7400 to grow to $39200 with a 6.25% annual interest rate using the continuous compound interest formula. This problem covers key concepts like exponential growth and compound interest, suitable for students in Grades 10-12.

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