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Question 10 (Multiple Choice Worth 1 point)
(Compound Interest and Geometric Sequences LC)

The equation, [tex][tex]$A = P\left(1+\frac{0.054}{2}\right)^{2t}$[/tex][/tex], represents the amount of money earned on a compound interest savings account with an annual interest rate of 5.4% compounded semiannually. If the initial investment is [tex]\[tex]$ 3,000[/tex], determine the amount in the account after 15 years. Round the answer to the nearest hundredths place.

A. [tex]\$[/tex] 3,164.19[/tex]
B. [tex]\[tex]$ 6,671.67[/tex]
C. [tex]\$[/tex] 4,473.81[/tex]
D. [tex]\$ 14,532.47[/tex]

Sagot :

GinnyAnswer
To determine the amount of money in the compound interest savings account after 15 years, we will follow these steps:

1. Identify the given variables:
- Initial investment ([tex]\(P\)[/tex]) = [tex]$3000 - Annual interest rate (\(r\)) = 5.4% (or 0.054 in decimal form) - Compounding frequency (\(n\)) = 2 (since it is compounded semiannually) - Number of years (\(t\)) = 15 2. Substitute these values into the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 3. Plug in the given values: \[ A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \times 15} \] 4. Calculate the periodic interest rate: \[ \frac{0.054}{2} = 0.027 \] 5. Calculate the exponent: \[ 2 \times 15 = 30 \] 6. Substitute these into the formula: \[ A = 3000 \left(1 + 0.027\right)^{30} \] 7. Calculate inside the parenthesis: \[ 1 + 0.027 = 1.027 \] 8. Raise 1.027 to the 30th power: \[ 1.027^{30} \] 9. Multiply this result by 3000: \[ 3000 \times \left(1.027^{30}\right) \] 10. The calculated amount \(A\) after 15 years, rounded to the nearest hundredths place, is: \[ \boxed{6671.67} \] Therefore, the amount in the account after 15 years is \( \$[/tex] 6,671.67 \). The correct answer is:
[tex]\[ \boxed{\$ 6,671.67} \][/tex]
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