Certainly! Let’s walk through the steps to find the value of Erica's investment after 6 years.
### Initial Setup
1. Initial Investment (Principal): Erica invested [tex]$4000 initially.
2. Annual Growth Rate: The value of her investment grows at an annual rate of 4%. This translates to a growth multiplier of \( 1 + 0.04 = 1.04 \).
3. Time (t): The investment duration is 6 years.
### Formula for Compound Interest
The formula to calculate the future value of an investment compounded annually is given by:
\[
y = P (1 + r)^t
\]
where:
- \( y \) is the future value of the investment.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual growth rate (expressed as a decimal).
- \( t \) is the time in years.
### Plugging in the Values
Using the values provided:
- \( P = 4000 \)
- \( r = 0.04 \) (which is 4% expressed as a decimal)
- \( t = 6 \)
Substitute these into the formula:
\[
y = 4000 (1 + 0.04)^6
\]
\[
y = 4000 (1.04)^6
\]
### Calculating the Growth Factor
Next, calculate \( (1.04)^6 \). Using a calculator for precision:
\[
(1.04)^6 = 1.265319
\]
### Calculating the Future Value
Now, multiply the initial investment by the growth factor:
\[
y = 4000 \times 1.265319
\]
Perform the multiplication:
\[
y = 5061.276
\]
### Conclusion
Therefore, the value of Erica's investment after 6 years will be approximately $[/tex]5061.28.
Erica’s investment grows to \$5061.28 over a period of 6 years at an annual growth rate of 4%.
