To determine how many years the account had been accumulating interest using the compound interest formula, we start with the given expression representing the value of the investment account:
[tex]\[ 10,350\left(1+\frac{0.04}{12}\right)^{120} \][/tex]
The compound interest formula is:
[tex]\[ A(t)=P\left(1+\frac{r}{n}\right)^{nt} \][/tex]
Here, the parameters are:
- [tex]\( P \)[/tex]: the principal amount ($10,350)
- [tex]\( r \)[/tex]: the annual interest rate (0.04)
- [tex]\( n \)[/tex]: the number of times the interest is compounded per year (12)
- [tex]\( t \)[/tex]: the number of years the money is invested or borrowed for.
In the given expression, the term inside the parenthesis is:
[tex]\[ \left(1+\frac{0.04}{12}\right) \][/tex]
And it is raised to the power of 120:
[tex]\[ \left(1+\frac{0.04}{12}\right)^{120} \][/tex]
In the compound interest formula, the exponent [tex]\( nt \)[/tex] represents the total number of compounding periods. Here, it is given as 120.
To find out the number of years [tex]\( t \)[/tex], we can use the relation:
[tex]\[ nt = 120 \][/tex]
Given:
[tex]\[ n = 12 \][/tex] (compounding periods per year)
So, we can solve for [tex]\( t \)[/tex]:
[tex]\[ 12t = 120 \][/tex]
Divide both sides of the equation by 12:
[tex]\[ t = \frac{120}{12} \][/tex]
Thus:
[tex]\[ t = 10 \][/tex]
Therefore, the account had been accumulating interest for:
[tex]\[ \boxed{10} \][/tex] years.
