To determine how much money will be in the account after 10 years, we need to use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money in the account after [tex]\( t \)[/tex] years.
- [tex]\( P \)[/tex] is the initial deposit (principal).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for.
Let's apply the given values to the formula:
1. Initial deposit ([tex]\( P \)[/tex]): [tex]$4000
2. Annual interest rate (\( r \)): 6%, which is 0.06 as a decimal.
3. Times compounded per year (\( n \)): monthly, so \( n = 12 \).
4. Number of years (\( t \)): 10
Substitute these values into the compound interest formula:
\[ A = 4000 \left(1 + \frac{0.06}{12}\right)^{12 \times 10} \]
First, calculate the rate per period:
\[ \frac{0.06}{12} = 0.005 \]
Then, add 1 to it:
\[ 1 + 0.005 = 1.005 \]
Now raise this value to the power of the total number of compounding periods (\( 12 \times 10 \)):
\[ (1.005)^{120} \]
Using a calculator, we find:
\[ (1.005)^{120} \approx 1.819396792 \]
Finally, multiply this result by the initial deposit (\( 4000 \)):
\[ 4000 \times 1.819396792 \approx 7277.59 \]
So, the amount in the account after 10 years will be approximately $[/tex]7277.59.
