How long will it take an investment to double in value if the interest rate is 11% compounded continuously? (Round your answer to two decimal places.)
 
What is the equivalent annual interest rate? (Round your answer to two decimal places.)

To solve these problems, we will use the formulas for continuous compounding and equivalent annual interest rate.

### 1. Time for Investment to Double (Continuous Compounding)

The formula for continuous compounding is:

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

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for (years).
- \( e \) is the base of the natural logarithm.

For the investment to double, \( A = 2P \). Substituting this into the formula gives:

\[ 2P = P e^{rt} \]

Dividing both sides by \( P \):

\[ 2 = e^{rt} \]

Taking the natural logarithm of both sides:

\[ \ln(2) = rt \]

Solving for \( t \):

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

Given \( r = 0.11 \):

\[ t = \frac{\ln(2)}{0.11} \]

Using a calculator:

\[ \ln(2) \approx 0.6931 \]

\[ t = \frac{0.6931}{0.11} \approx 6.30 \]

So, it will take approximately 6.30 years for the investment to double.

### 2. Equivalent Annual Interest Rate

The equivalent annual interest rate (EAR) for continuous compounding is calculated using the formula:

\[ \text{EAR} = e^r - 1 \]

Given \( r = 0.11 \):

\[ \text{EAR} = e^{0.11} - 1 \]

Using a calculator:

\[ e^{0.11} \approx 1.1163 \]

\[ \text{EAR} = 1.1163 - 1 \approx 0.1163 \]

Converting to a percentage:

\[ \text{EAR} \approx 11.63\% \]

So, the equivalent annual interest rate is approximately 11.63%.

### Summary

1. **Time for Investment to Double**: \( \approx 6.30 \) years.
2. **Equivalent Annual Interest Rate**: \( \approx 11.63\% \).

Would you like further details or have any questions?

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Here are 8 related questions for further exploration:
1. How does the continuous compounding formula change if the interest rate varies annually?
2. What would the doubling time be if the interest rate were 8% instead of 11%?
3. How does the concept of continuous compounding differ from compounding annually or quarterly?
4. Can the continuous compounding formula be applied to depreciation? If so, how?
5. How does the natural logarithm function simplify solving for time in continuous compounding problems?
6. What are the practical applications of continuous compounding in finance and economics?
7. How would you derive the formula for the equivalent annual interest rate starting from continuous compounding principles?
8. What is the impact of inflation on continuously compounded investments over time?

**Tip:** Always double-check calculations, especially when using exponential and logarithmic functions, to ensure accuracy in financial mathematics.

Learn how to calculate the time it takes for an investment to double with an 11% continuous interest rate and find its equivalent annual interest rate. Detailed step-by-step solution using continuous compounding and logarithmic functions.

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