Certainly! Let's solve the problem step by step using the formula for compound interest:
Given:
- [tex]\( P = \$ 5,000 \)[/tex] (the principal amount or initial investment)
- [tex]\( r = 0.02 \)[/tex] (the annual interest rate, expressed as a decimal)
- [tex]\( m = 4 \)[/tex] (the number of times interest is compounded per year, quarterly)
- [tex]\( t = 10 \)[/tex] (the number of years)
The formula for the future value with compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{m}\right)^{mt} \][/tex]
Let's plug these values into the formula:
1. Calculate [tex]\( \frac{r}{m} \)[/tex]:
[tex]\[
\frac{r}{m} = \frac{0.02}{4} = 0.005
\][/tex]
2. Add 1 to [tex]\( \frac{r}{m} \)[/tex]:
[tex]\[
1 + \frac{r}{m} = 1 + 0.005 = 1.005
\][/tex]
3. Calculate [tex]\( mt \)[/tex]:
[tex]\[
mt = 4 \times 10 = 40
\][/tex]
4. Raise [tex]\( 1.005 \)[/tex] to the power of [tex]\( 40 \)[/tex]:
[tex]\[
(1.005)^{40}
\][/tex]
5. Multiply the result by the principal amount [tex]\( P \)[/tex]:
[tex]\[
A = \$5,000 \times (1.005)^{40}
\][/tex]
After calculating the above expression, you find that:
[tex]\[
A \approx \$ 6,103.97
\][/tex]
Therefore, the amount of money in the savings account after 10 years is approximately [tex]\(\$ 6,103.97\)[/tex].
The correct answer is:
[tex]\[
\$ 6,103.97
\][/tex]
