If an amount doubles in 8 years with continuous compounding, what is the annual interest rate 

To find the annual interest rate when an amount doubles in 8 years with continuous compounding, we can use the continuous compounding formula:

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

Where:
- \(A\) is the final amount,
- \(P\) is the initial amount,
- \(r\) is the annual interest rate,
- \(t\) is the time in years,
- \(e\) is Euler's number (approximately 2.71828).

Since the amount doubles, \(A = 2P\). We also know that \(t = 8\) years. Substituting these into the formula:

\[
2P = P e^{8r}
\]

Dividing both sides by \(P\):

\[
2 = e^{8r}
\]

Taking the natural logarithm (ln) of both sides to eliminate \(e\):

\[
\ln(2) = 8r
\]

Solving for \(r\):

\[
r = \frac{\ln(2)}{8}
\]

Using \(\ln(2) \approx 0.6931\):

\[
r = \frac{0.6931}{8} \approx 0.08664
\]

Thus, the annual interest rate is approximately 0.08664, or **8.66%**.

Would you like more details or have any questions about this?

Here are 5 related questions:
1. How would the formula change if the interest was compounded monthly instead of continuously?
2. What would the interest rate be if the amount tripled in 8 years instead of doubling?
3. Can the continuous compounding formula be applied to any type of investment or savings plan?
4. How long would it take for the amount to double if the interest rate was 5%?
5. What is the difference between continuous compounding and simple interest?

**Tip:** Continuous compounding assumes that interest is being added at every possible moment, providing the maximum possible growth rate for a given interest rate.

If an amount doubles in 8 years with continuous compounding, what is the annual interest rate?

Discover how to calculate the annual interest rate when an amount doubles in 8 years with continuous compounding. Learn to apply the continuous compounding formula A = Pe^{rt}, use natural logarithms, and solve for the interest rate. Suitable for high school students in Grades 10-12.

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