Let's solve the problem step-by-step using the provided formulas.
1. Compounded Semi-Annually:
The formula for compound interest is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Here, the principal amount [tex]\( P = 15,000 \)[/tex] dollars, annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex], time [tex]\( t = 4 \)[/tex] years, and interest is compounded semi-annually. For semi-annual compounding, [tex]\( n = 2 \)[/tex] (since interest is compounded twice a year).
Substitute the values into the formula:
[tex]\[
A = 15,000 \left(1 + \frac{0.065}{2}\right)^{2 \times 4}
\][/tex]
[tex]\[
A = 15,000 \left(1 + 0.0325\right)^{8}
\][/tex]
[tex]\[
A \approx 15,000 \times 1.291577
\][/tex]
[tex]\[
A \approx 19,373.66
\][/tex]
2. Compounded Quarterly:
For quarterly compounding, [tex]\( n = 4 \)[/tex] (since interest is compounded four times a year).
Substitute the values into the formula:
[tex]\[
A = 15,000 \left(1 + \frac{0.065}{4}\right)^{4 \times 4}
\][/tex]
[tex]\[
A = 15,000 \left(1 + 0.01625\right)^{16}
\][/tex]
[tex]\[
A \approx 15,000 \times 1.294222
\][/tex]
[tex]\[
A \approx 19,413.34
\][/tex]
3. Compounded Monthly:
For monthly compounding, [tex]\( n = 12 \)[/tex] (since interest is compounded twelve times a year).
Substitute the values into the formula:
[tex]\[
A = 15,000 \left(1 + \frac{0.065}{12}\right)^{12 \times 4}
\][/tex]
[tex]\[
A = 15,000 \left(1 + 0.0054167\right)^{48}
\][/tex]
[tex]\[
A \approx 15,000 \times 1.296021
\][/tex]
[tex]\[
A \approx 19,440.31
\][/tex]
4. Compounded Continuously:
The formula for continuous compounding is:
[tex]\[
A = P e^{rt}
\][/tex]
Substitute the values into the formula:
[tex]\[
A = 15,000 \times e^{0.065 \times 4}
\][/tex]
[tex]\[
A = 15,000 \times e^{0.26}
\][/tex]
Approximate [tex]\( e^{0.26} \)[/tex]:
[tex]\[
A \approx 15,000 \times 1.296930
\][/tex]
[tex]\[
A \approx 19,453.95
\][/tex]
Here are the accumulated values:
a. Compounded semi-annually: [tex]\( A \approx 19,373.66 \)[/tex]
b. Compounded quarterly: [tex]\( A \approx 19,413.34 \)[/tex]
c. Compounded monthly: [tex]\( A \approx 19,440.31 \)[/tex]
d. Compounded continuously: [tex]\( A \approx 19,453.95 \)[/tex]
Thus, when the money is compounded continuously, the accumulated value is 19,453.95.
