Certainly! Let's solve this step-by-step.
### Part (a): Using the Future Value Formula
The future value (FV) formula for annual compounding is expressed as:
[tex]\[ \text{FV} = P \times (1 + r)^t \][/tex]
Where:
- [tex]\( \text{FV} \)[/tex] is the future value of the investment.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the annual interest rate.
- [tex]\( t \)[/tex] is the time in years.
Given:
- [tex]\( P = \$1,600 \)[/tex]
- [tex]\( r = 0.86 \)[/tex] (which is 86% expressed as a decimal)
- [tex]\( \text{FV} = 2 \times P = 2 \times 1,600 = \$3,200 \)[/tex] (since we want to find the time it takes to double the investment)
Rearranging the formula to solve for [tex]\( t \)[/tex], we get:
[tex]\[ 2P = P \times (1 + r)^t \][/tex]
[tex]\[ 2 = (1 + r)^t \][/tex]
[tex]\[ 2 = (1 + 0.86)^t \][/tex]
[tex]\[ 2 = 1.86^t \][/tex]
Next, take the natural logarithm (logarithm base [tex]\( e \)[/tex]) of both sides:
[tex]\[ \ln(2) = \ln(1.86^t) \][/tex]
Using the logarithmic property [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]:
[tex]\[ \ln(2) = t \cdot \ln(1.86) \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(2)}{\ln(1.86)} \][/tex]
Approximately calculating [tex]\( t \)[/tex]:
[tex]\[ t \approx 1.12 \text{ years} \][/tex]
#### Answer for Part (a):
Based on the future value formula, it will take approximately 1.12 years for her investment to double.
### Part (b): Using the Rule of 72
The Rule of 72 is a simplified way to estimate the number of years required to double the investment at a given annual rate of return. It is given by:
[tex]\[ t = \frac{72}{r \cdot 100} \][/tex]
Where:
- [tex]\( r \)[/tex] is the annual interest rate in percentage form.
Given:
- [tex]\( r = 0.86 \times 100 = 86\% \)[/tex]
So, we have:
[tex]\[ t = \frac{72}{86} \][/tex]
Approximately calculating [tex]\( t \)[/tex]:
[tex]\[ t \approx 0.84 \text{ years} \][/tex]
#### Answer for Part (b):
Using the Rule of 72, it will take approximately 0.84 years for her investment to double.
### Conclusion
- Part (a): Future Value Formula - 1.12 years
- Part (b): Rule of 72 - 0.84 years
