Makahanap ng eksaktong solusyon sa iyong mga problema sa IDNStudy.com. Ang aming mga eksperto ay nagbibigay ng mabilis at eksaktong sagot upang tulungan kang maunawaan at malutas ang anumang problema.
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%.