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Sagot :
Let's address each part of the question one by one, using the given results:
### a. Semiannually Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 2 \)[/tex] (since it's semiannual)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{2} \right)^{2 \times 4} \][/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]\[ \$ 19373.66 \][/tex]
### b. Quarterly Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 4 \)[/tex] (since it's quarterly)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{4} \right)^{4 \times 4} \][/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]\[ \$ 19413.34 \][/tex]
### c. Monthly Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 12 \)[/tex] (since it's monthly)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{12} \right)^{12 \times 4} \][/tex]
[tex]\[ A = 15000 \left( 1 + 0.00541667 \right)^{48} \][/tex]
[tex]\[ A = 15000 \left( 1.00541667 \right)^{48} \][/tex]
[tex]\[ A \approx 19440.31 \][/tex]
So, the accumulated value if the money is compounded monthly is:
[tex]\[ \$ 19440.31 \][/tex]
### Summarizing the Results:
a. Semiannually Compounded Interest: [tex]\(\$ 19373.66\)[/tex]
b. Quarterly Compounded Interest: [tex]\(\$ 19413.34\)[/tex]
c. Monthly Compounded Interest: [tex]\(\$ 19440.31\)[/tex]
### a. Semiannually Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 2 \)[/tex] (since it's semiannual)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{2} \right)^{2 \times 4} \][/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]\[ \$ 19373.66 \][/tex]
### b. Quarterly Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 4 \)[/tex] (since it's quarterly)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{4} \right)^{4 \times 4} \][/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]\[ \$ 19413.34 \][/tex]
### c. Monthly Compounded Interest
Given:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 6.5\% = 0.065 \)[/tex]
- Time period [tex]\( t = 4 \)[/tex] years
- Compounding frequency [tex]\( n = 12 \)[/tex] (since it's monthly)
The compound interest formula is:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Plugging in the values:
[tex]\[ A = 15000 \left( 1 + \frac{0.065}{12} \right)^{12 \times 4} \][/tex]
[tex]\[ A = 15000 \left( 1 + 0.00541667 \right)^{48} \][/tex]
[tex]\[ A = 15000 \left( 1.00541667 \right)^{48} \][/tex]
[tex]\[ A \approx 19440.31 \][/tex]
So, the accumulated value if the money is compounded monthly is:
[tex]\[ \$ 19440.31 \][/tex]
### Summarizing the Results:
a. Semiannually Compounded Interest: [tex]\(\$ 19373.66\)[/tex]
b. Quarterly Compounded Interest: [tex]\(\$ 19413.34\)[/tex]
c. Monthly Compounded Interest: [tex]\(\$ 19440.31\)[/tex]
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