Answer:
Step-by-step explanation: Let's solve the problem step-by-step for both parts using the given methods:
**A) Using the Future Value Formula:**
The future value formula for compound interest is:
\[ A = P \left(1 + \frac{r}{m}\right)^{mt} \]
where:
- \( A \) is the future value of the investment,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( m \) is the number of compounding periods per year,
- \( t \) is the number of years.
In this case:
- \( P = 1600 \) dollars,
- \( r = 8.6\% = 0.086 \) (decimal form),
- We want \( A = 3200 \) dollars (double the initial investment),
- Compounded annually (\( m = 1 \)).
Let's solve for \( t \):
\[ 3200 = 1600 \left(1 + \frac{0.086}{1}\right)^{1 \cdot t} \]
Divide both sides by 1600:
\[ 2 = \left(1 + 0.086\right)^t \]
Take the natural logarithm (ln) of both sides to solve for \( t \):
\[ \ln(2) = t \cdot \ln(1.086) \]
Now, solve for \( t \):
\[ t = \frac{\ln(2)}{\ln(1.086)} \]
Using a calculator:
\[ t \approx \frac{0.693147}{0.083381} \approx 8.32 \]
So, using the future value formula, it will take approximately **8.32 years** for her investment to double.
**B) Using the Rule of 72:**
The Rule of 72 states that you can estimate the number of years it takes for an investment to double by dividing 72 by the annual interest rate (as a percentage):
\[ \text{Years to double} = \frac{72}{\text{annual interest rate}} \]
In this case:
\[ \text{Years to double} = \frac{72}{8.6} \approx 8.37 \]
Rounded to two decimal places, using the Rule of 72, it will take approximately **8.37 years** for her investment to double.
### Summary:
- **A)** Using the future value formula: Approximately **8.32 years**.
- **B)** Using the Rule of 72: Approximately **8.37 years**.
These calculations provide two different methods to estimate the time it takes for an investment to double under annual compounding with an interest rate of 8.6%.
