Absolutely, let's solve this step-by-step using the compound interest formula to find out the annual interest rate.
The compound interest formula is:
[tex]\[ V(t) = P \left(1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( P = 2500 \)[/tex] (initial principal investment, in dollars)
- [tex]\( V(t) = 2762.35 \)[/tex] (value of the investment after [tex]\( t \)[/tex] years, in dollars)
- [tex]\( t = 2 \)[/tex] (years since initial deposit)
- [tex]\( n = 12 \)[/tex] (number of times interest is compounded per year)
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal)
To find the annual interest rate ([tex]\( r \)[/tex]) we need to rearrange the formula. Start by isolating [tex]\( \frac{r}{n} \)[/tex]:
[tex]\[ 2762.35 = 2500 \left(1 + \frac{r}{12} \right)^{12 \cdot 2} \][/tex]
Divide both sides by 2500:
[tex]\[ \frac{2762.35}{2500} = \left(1 + \frac{r}{12} \right)^{24} \][/tex]
Calculate the left-hand side:
[tex]\[ 1.10494 = \left(1 + \frac{r}{12} \right)^{24} \][/tex]
To solve for [tex]\( r \)[/tex], we take the 24th root of both sides:
[tex]\[ \left(1.10494 \right)^{\frac{1}{24}} = 1 + \frac{r}{12} \][/tex]
Calculate the 24th root:
[tex]\[ 1.004166 \approx 1 + \frac{r}{12} \][/tex]
Subtract 1 from both sides to isolate [tex]\( \frac{r}{12} \)[/tex]:
[tex]\[ \frac{r}{12} \approx 0.004166 \][/tex]
Multiply both sides by 12 to solve for [tex]\( r \)[/tex]:
[tex]\[ r \approx 0.004166 \times 12 \][/tex]
[tex]\[ r \approx 0.049999 \][/tex]
Convert [tex]\( r \)[/tex] to a percentage by multiplying by 100:
[tex]\[ r \approx 4.9999\% \][/tex]
Therefore, the annual interest rate for the account is approximately [tex]\( 5\% \)[/tex]. The correct answer from the given options is:
[tex]\[ \boxed{5\%} \][/tex]
