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Sagot :
To solve the compound interest problem for [tex]$3,000 at 8% interest for 5 years, let's compute the total amounts and the interest amounts for different compounding frequencies: annually, semiannually, and quarterly.
### Annually Compounded Interest
1. Total Amount: The formula for compound interest is given by:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
For annual compounding:
- \( P = 3000 \) (principal)
- \( r = 0.08 \) (annual interest rate)
- \( n = 1 \) (compounding frequency per year)
- \( t = 5 \) (time in years)
Plugging in these values:
\[
A = 3000 \left(1 + \frac{0.08}{1}\right)^{1 \times 5} = 3000 \left(1 + 0.08\right)^{5} = 3000 \times 1.4693 \approx 4407.98
\]
2. Interest Amount: The interest amount is the total amount minus the principal:
\[
\text{Interest} = A - P = 4407.98 - 3000 \approx 1407.98
\]
### Semiannually Compounded Interest
1. Total Amount: For semiannual compounding:
- \( n = 2 \)
Plugging in the values:
\[
A = 3000 \left(1 + \frac{0.08}{2}\right)^{2 \times 5} = 3000 \left(1 + 0.04\right)^{10} = 3000 \times 1.48 \approx 4440.73
\]
2. Interest Amount:
\[
\text{Interest} = 4440.73 - 3000 \approx 1440.73
\]
### Quarterly Compounded Interest
1. Total Amount: For quarterly compounding:
- \( n = 4 \)
Plugging in the values:
\[
A = 3000 \left(1 + \frac{0.08}{4}\right)^{4 \times 5} = 3000 \left(1 + 0.02\right)^{20} = 3000 \times 1.486 \approx 4457.84
\]
2. Interest Amount:
\[
\text{Interest} = 4457.84 - 3000 \approx 1457.84
\]
### Summary Table
Let's fill in your summary table with the calculated values:
\[
\begin{tabular}{|l|l|l|}
\hline
Compounding & Total Amount & Interest Amount \\
\hline
annually & \(\$[/tex]4407.98\) & [tex]\(\$1407.98\)[/tex] \\
\hline
semiannually & [tex]\(\$4440.73\)[/tex] & [tex]\(\$1440.73\)[/tex] \\
\hline
quarterly & [tex]\(\$4457.84\)[/tex] & [tex]\(\$1457.84\)[/tex] \\
\hline
\end{tabular}
\]
Thus, the total amounts and interest amounts for each compounding period are as listed in the table above.
\hline
semiannually & [tex]\(\$4440.73\)[/tex] & [tex]\(\$1440.73\)[/tex] \\
\hline
quarterly & [tex]\(\$4457.84\)[/tex] & [tex]\(\$1457.84\)[/tex] \\
\hline
\end{tabular}
\]
Thus, the total amounts and interest amounts for each compounding period are as listed in the table above.
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