Answered

The equation [tex]A = P \left(1 + \frac{0.054}{2}\right)^{2t}[/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]$3,000, determine the amount in the account after 15 years. Round the answer to the nearest hundredth.

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



Answer :

GinnyAnswer
Let's analyze the problem step-by-step to find the amount of money that will be in the account after 15 years through compound interest.

### Step-by-Step Solution:

1. Identify the given values:
- Initial investment ([tex]\(P\)[/tex]): [tex]$3000 - Annual interest rate (\(r\)): 5.4%, which can be expressed as 0.054 when converted to decimal form. - Number of times the interest is compounded per year (\(n\)): 2 (since it is compounded semiannually) - Time (\(t\)): 15 years 2. Use the compound interest formula: The formula to calculate the amount of money accumulated, \(A\), using compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 3. Substitute the given values into the formula: \[ A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \times 15} \] 4. Simplify the expression inside the parentheses: \[ \left(1 + \frac{0.054}{2}\right) = \left(1 + 0.027\right) = 1.027 \] 5. Calculate the exponent \(2t\): \[ 2 \times 15 = 30 \] 6. Raise the base to the power of 30: \[ (1.027)^{30} \] 7. Multiply the initial investment by this result: \[ A = 3000 \times (1.027)^{30} \] 8. Round the final amount to the nearest hundredth: \[ A \approx 6671.67 \] Therefore, the amount in the account after 15 years, rounded to the nearest hundredth, is: \[ \$[/tex]6671.67
\]

Given the multiple-choice answers:
- [tex]$\$[/tex]3164.19[tex]$ - \$[/tex]6671.67[tex]$ - $[/tex]\[tex]$4473.81$[/tex]
- [tex]$\$[/tex]14532.47[tex]$ The correct answer is \(\$[/tex]6671.67\).

Other Questions