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].
