To find the future value of the investment and the interest earned, we use the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Where:
- [tex]\( P \)[/tex] is the principal investment, which is [tex]\( \$45,000 \)[/tex].
- [tex]\( r \)[/tex] is the annual rate of interest in decimal form, which is [tex]\( 0.03 \)[/tex] (3%).
- [tex]\( n \)[/tex] is the number of compounding periods per year, which is [tex]\( 12 \)[/tex] (monthly).
- [tex]\( t \)[/tex] is the number of years the money is invested, which is [tex]\( 5 \)[/tex] years.
Substituting the given values into the formula, we get:
[tex]\[
A = 45000 \left(1 + \frac{0.03}{12}\right)^{12 \cdot 5}
\][/tex]
First, calculate the monthly interest rate:
[tex]\[
\frac{0.03}{12} = 0.0025
\][/tex]
Next, calculate the exponent:
[tex]\[
12 \times 5 = 60
\][/tex]
Thus, the formula becomes:
[tex]\[
A = 45000 \left(1 + 0.0025\right)^{60}
\][/tex]
Calculate [tex]\( 1 + 0.0025 \)[/tex]:
[tex]\[
1 + 0.0025 = 1.0025
\][/tex]
Now raise [tex]\( 1.0025 \)[/tex] to the power of [tex]\( 60 \)[/tex]:
Evaluating [tex]\( 1.0025^{60} \)[/tex] yields approximately [tex]\( 1.1638396 \)[/tex].
Now multiply this by the principal:
[tex]\[
A = 45000 \times 1.1638396 \approx 52272.76
\][/tex]
So the future value of the account is:
[tex]\[
\text{Future Value} = \$52,272.76
\][/tex]
To find the interest earned, we subtract the principal from the future value:
[tex]\[
\text{Interest} = A - P = 52272.76 - 45000 = 7272.76
\][/tex]
So the interest earned in the account over 5 years is:
[tex]\[
\text{Interest} = \$7,272.76
\][/tex]
In conclusion:
- The future value of the account is [tex]\( \$52,272.76 \)[/tex].
- The interest earned over 5 years is [tex]\( \$7,272.76 \)[/tex].
