Sure, let's solve for each of the three scenarios using the compound interest formulas provided.
### a. Accumulated Value with Semiannual Compounding
For semiannual compounding, we use the formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Here:
- [tex]\( P = 15000 \)[/tex] (the principal amount)
- [tex]\( r = 0.065 \)[/tex] (the annual interest rate in decimal form)
- [tex]\( n = 2 \)[/tex] (number of compounding periods per year)
- [tex]\( t = 4 \)[/tex] (time in years)
Substitute the values into the formula:
[tex]\[ A = 15000 \left(1 + \frac{0.065}{2}\right)^{2 \cdot 4} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.0325\right)^{8} \][/tex]
[tex]\[ A = 15000 \left(1.0325\right)^{8} \][/tex]
[tex]\[ A \approx 19373.66 \][/tex]
So, the accumulated value if the money is compounded semiannually is:
[tex]\[ \boxed{19373.66} \][/tex]
### b. Accumulated Value with Quarterly Compounding
For quarterly compounding, we use the same formula with:
- [tex]\( P = 15000 \)[/tex]
- [tex]\( r = 0.065 \)[/tex]
- [tex]\( n = 4 \)[/tex]
- [tex]\( t = 4 \)[/tex]
Substitute the values into the formula:
[tex]\[ A = 15000 \left(1 + \frac{0.065}{4}\right)^{4 \cdot 4} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.01625\right)^{16} \][/tex]
[tex]\[ A = 15000 \left(1.01625\right)^{16} \][/tex]
[tex]\[ A \approx 19413.34 \][/tex]
So, the accumulated value if the money is compounded quarterly is:
[tex]\[ \boxed{19413.34} \][/tex]
### d. Accumulated Value with Continuous Compounding
For continuous compounding, we use the formula:
[tex]\[ A = P e^{rt} \][/tex]
Here:
- [tex]\( P = 15000 \)[/tex]
- [tex]\( r = 0.065 \)[/tex]
- [tex]\( t = 4 \)[/tex]
Substitute the values into the formula:
[tex]\[ A = 15000 e^{0.065 \cdot 4} \][/tex]
[tex]\[ A = 15000 e^{0.26} \][/tex]
[tex]\[ A \approx 15000 \cdot 1.29693008666577 \][/tex]
[tex]\[ A \approx 19453.95 \][/tex]
So, the accumulated value if the money is compounded continuously is:
[tex]\[ \boxed{19453.95} \][/tex]
