Let's solve the problem step by step:
### Step 1: Understand the Problem
We're asked to determine the amount of money that will be in an account after 7 years given:
- Initial investment (principal) = \[tex]$8,000
- Annual interest rate = 3.5% (0.035 as a decimal)
- Compounding periods = Quarterly (4 times a year)
- Time = 7 years
### Step 2: Use the Compound Interest Formula
The compound interest formula is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times interest is compounded per year.
- \( t \) is the number of years the money is invested for.
### Step 3: Substitute the Given Values into the Formula
From our given problem:
- \( P = 8000 \)
- \( r = 0.035 \)
- \( n = 4 \)
- \( t = 7 \)
Substitute these values into the compound interest formula:
\[ A = 8000 \left(1 + \frac{0.035}{4}\right)^{4 \times 7} \]
### Step 4: Calculate the Intermediate Steps
First, calculate the term inside the parentheses:
\[ \frac{0.035}{4} = 0.00875 \]
So the expression becomes:
\[ 1 + 0.00875 = 1.00875 \]
Now calculate the exponent part:
\[ 4 \times 7 = 28 \]
So the formula now looks like:
\[ A = 8000 \left(1.00875\right)^{28} \]
### Step 5: Compute the Exponent
Raise \( 1.00875 \) to the power of 28:
\[ 1.00875^{28} \approx 1.2712 \] (Value approximated for simplicity)
### Step 6: Multiply by the Principal
Finally, multiply this result by the principal (\$[/tex]8,000):
[tex]\[ A = 8000 \times 1.2712 \approx 10169.6 \][/tex]
### Step 7: Round Off if Necessary
If we round this to the nearest cent, the amount in the account after 7 years will be approximately:
[tex]\[ A \approx \$10,169.60 \][/tex]
### Answer
After 7 years, the amount of money in your account will be approximately \$10,169.60.
