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Use the compound interest formulas [tex]$A=P\left(1+\frac{r}{n}\right)^{nt}$[/tex] and [tex]$A=Pe^{rt}$[/tex] to solve the problem given. Round answers to the nearest cent.

Find the accumulated value of an investment of [tex][tex]$\$[/tex]15,000[/tex] for 4 years at an interest rate of [tex]6.5\%[/tex] if the money is:
a. Compounded semiannually
b. Compounded quarterly
c. Compounded monthly
d. Compounded continuously

a. What is the accumulated value if the money is compounded semiannually?
[tex]\[ \$ 19373.66 \][/tex]
(Round your answer to the nearest cent. Do not include the [tex]$\$[/tex] symbol in your answer.)

b. What is the accumulated value if the money is compounded quarterly?
[tex]\[ \$ 19,413.34 \][/tex]
(Round your answer to the nearest cent. Do not include the [tex]$\$[/tex] symbol in your answer.)

c. What is the accumulated value if the money is compounded monthly?
[tex]\[ \$ \square \][/tex]
(Round your answer to the nearest cent. Do not include the [tex]$\$[/tex] symbol in your answer.)

d. What is the accumulated value if the money is compounded continuously?
[tex]\[ \$ \square \][/tex]
(Round your answer to the nearest cent. Do not include the [tex]$\$[/tex] symbol in your answer.)

Sagot :

GinnyAnsweridn
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]
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