To find out how much money needs to be deposited today to have [tex]$3,500 in 11 years with an annual interest rate of 5.5% compounded continuously, we need to use the formula for continuously compounded interest. Here is a step-by-step solution:
1. Understand the formula:
The formula for continuous compounding is given by:
\[
PV = \frac{FV}{e^{rt}}
\]
where:
- \( PV \) is the present value (initial investment),
- \( FV \) is the future value,
- \( e \) is the base of the natural logarithm (approximately 2.71828),
- \( r \) is the annual interest rate (expressed as a decimal),
- \( t \) is the time in years.
2. Identify the given values:
- Future Value (\( FV \)) = $[/tex]3,500
- Annual interest rate ([tex]\( r \)[/tex]) = 5.5% = 0.055
- Time ([tex]\( t \)[/tex]) = 11 years
3. Substitute the values into the formula:
[tex]\[
PV = \frac{3500}{e^{0.055 \times 11}}
\][/tex]
4. Calculate [tex]\( e^{0.055 \times 11} \)[/tex]:
[tex]\[
e^{0.605} \approx 1.83156388887
\][/tex]
5. Divide the future value by [tex]\( e^{0.605} \)[/tex]:
[tex]\[
PV = \frac{3500}{1.83156388887} \approx 1911.260493238983
\][/tex]
6. Round the result to 2 decimal places:
[tex]\[
PV \approx 1911.26
\][/tex]
So, the initial investment required to reach a future value of [tex]$3,500 in 11 years with a continuous compounding interest rate of 5.5% per year is \(\$[/tex]1911.26\).
