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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 place.

A. \$[/tex]3,164.19
B. \[tex]$26,671.67
C. \$[/tex]24,473.81
D. \$14,532.47



Answer :

GinnyAnswer
Sure, I'd be happy to help guide you through the compound interest calculations step-by-step.

Let's break down the problem and walk through the steps together:

### Step 1: Understanding the Formula
The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] = the amount of money accumulated after [tex]\( n \)[/tex] years, including interest.
- [tex]\( P \)[/tex] = the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] = the annual interest rate (decimal).
- [tex]\( n \)[/tex] = the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] = the time the money is invested for in years.

### Step 2: Given Values
From the question:
- [tex]\( P = 3000 \)[/tex] dollars
- [tex]\( r = 0.054 \)[/tex] (annual interest rate of [tex]\( 5.4\% \)[/tex])
- [tex]\( n = 2 \)[/tex] (compounded semiannually)
- [tex]\( t = 15 \)[/tex] years

### Step 3: Plugging Values into the Formula
We need to substitute the given values into our formula:
[tex]\[ A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \cdot 15} \][/tex]

### Step 4: Simplifying Inside the Parentheses
First, calculate [tex]\( \frac{0.054}{2} \)[/tex]:
[tex]\[ \frac{0.054}{2} = 0.027 \][/tex]

So, the equation becomes:
[tex]\[ A = 3000 \left(1 + 0.027\right)^{30} \][/tex]

### Step 5: Calculating Inside the Parentheses
Add 1 to 0.027:
[tex]\[ 1 + 0.027 = 1.027 \][/tex]

### Step 6: Raising to the Power
Calculate [tex]\( \left(1.027\right)^{30} \)[/tex]:
[tex]\[ (1.027)^{30} \approx 2.22389 \][/tex]

### Step 7: Final Multiplication
Multiply by the principal amount:
[tex]\[ A = 3000 \times 2.22389 \][/tex]

### Step 8: Calculation
Perform the multiplication:
[tex]\[ A \approx 3000 \times 2.22389 \approx 6671.67 \][/tex]

### Conclusion
Therefore, the amount in the savings account after 15 years, rounded to the nearest hundredth, is [tex]\( \$6671.67 \)[/tex].

So, the correct answer from the options provided is [tex]\( \$6671.67 \)[/tex].

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