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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 following problem.

Find the accumulated value of an investment of [tex] \$10,000 [/tex] for 3 years at an interest rate of [tex] 6.5\% [/tex]:

a. What is the accumulated value if the money is compounded semiannually?
[tex] \$12,115.47 [/tex] (Round your answer to the nearest cent.)

b. What is the accumulated value if the money is compounded monthly?
[tex] \$\square [/tex] (Round your answer to the nearest cent.)

c. What is the accumulated value if the money is compounded continuously?
[tex] \$\square [/tex] (Round your answer to the nearest cent.)


Sagot :

To solve this compound interest problem, we will start with part a, where the interest is compounded semiannually, and then proceed to part b, where the interest is compounded monthly.

### Part (a): Compounded Semiannually

1. Given Data:

- Principal ([tex]\( P \)[/tex]): \[tex]$10,000 - Annual Interest Rate (\( r \)): 0.065 (6.5%) - Time (\( t \)): 3 years - Compounding Periods per Year (\( n \)): 2 (since it's semiannually) 2. Formula for Compound Interest: \[ A = P \left(1+\frac{r}{n}\right)^{ nt } \] 3. Plugging in the Values: \[ A = 10,000 \left( 1 + \frac{0.065}{2} \right)^{2 \cdot 3} \] 4. Simplifying: \[ A = 10,000 \left( 1 + 0.0325 \right)^6 \] \[ A = 10,000 \left( 1.0325 \right)^6 \] 5. Calculating the Accumulated Value: \[ A \approx 12,115.47 \] Therefore, the accumulated value if compounded semiannually is \(\$[/tex] 12,115.47\).

### Part (b): Compounded Monthly

1. Given Data:

- Principal ( [tex]\( P \)[/tex]): \[tex]$10,000 - Annual Interest Rate (\( r \)): 0.065 (6.5%) - Time (\( t \)): 3 years - Compounding Periods per Year (\( n \)): 12 (since it's monthly) 2. Formula for Compound Interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{n t} \] 3. Plugging in the Values: \[ A = 10,000 \left( 1 + \frac{0.065}{12} \right)^{12 \cdot 3} \] 4. Simplifying: \[ A = 10,000 \left( 1 + 0.0054167 \right)^{36} \] \[ A = 10,000 \left( 1.0054167 \right)^{36} \] 5. Calculating the Accumulated Value: \[ A \approx 12,146.72 \] Therefore, the accumulated value if compounded monthly is \(\$[/tex] 12,146.72\).

By following the steps above, we've found the accumulated value for both semiannual and monthly compounding. Now let's fill in the answer:

### Final Answer for Part (b):

[tex]\[ \boxed{12,146.72} \][/tex]