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Answer:
To determine the interest rate, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the future value of the investment (3,875)
- \( P \) is the principal amount (2,050)
- \( r \) is the annual interest rate (to be found)
- \( n \) is the number of times interest is compounded per year (semi-annually, so \( n = 2 \))
- \( t \) is the time the money is invested for in years (4.5 years)
Plugging in the values:
\[ 3,875 = 2,050 \left(1 + \frac{r}{2}\right)^{2 \times 4.5} \]
This simplifies to:
\[ 3,875 = 2,050 \left(1 + \frac{r}{2}\right)^9 \]
Dividing both sides by 2,050:
\[ \left(1 + \frac{r}{2}\right)^9 = \frac{3,875}{2,050} \]
\[ \left(1 + \frac{r}{2}\right)^9 = 1.8902 \]
To solve for \( r \), take the 9th root of both sides:
\[ 1 + \frac{r}{2} = \left(1.8902\right)^{\frac{1}{9}} \]
Calculating the 9th root of 1.8902:
\[ 1 + \frac{r}{2} \approx 1.0747 \]
Subtracting 1 from both sides:
\[ \frac{r}{2} \approx 0.0747 \]
Multiplying both sides by 2:
\[ r \approx 0.1494 \]
Converting to a percentage and rounding to the nearest hundredth:
\[ r \approx 14.94\% \]
So, the annual interest rate is approximately 14.94%.