Continuous Interest Fa4967

1. **State the problem:** We need to find the interest earned on a principal amount of $30000$ deposited for $6$ months at an interest rate of $1 \frac{1}{2} \%$ per year, compounded continuously. 2. **Formula used:** For continuous compounding, the amount $A$ after time $t$ years is given by: $$A = P e^{rt}$$ where: - $P$ is the principal - $r$ is the annual interest rate (in decimal) - $t$ is the time in years - $e$ is Euler's number (approximately $2.71828$) 3. **Convert given values:** - Principal $P = 30000$ - Interest rate $r = 1 \frac{1}{2} \% = 1.5\% = 0.015$ - Time $t = 6$ months $= \frac{6}{12} = 0.5$ years 4. **Calculate the amount $A$:** $$A = 30000 \times e^{0.015 \times 0.5} = 30000 \times e^{0.0075}$$ 5. **Evaluate $e^{0.0075}$:** Using a calculator or approximation, $$e^{0.0075} \approx 1.00753$$ 6. **Calculate $A$:** $$A = 30000 \times 1.00753 = 30225.9$$ 7. **Find the interest earned:** Interest $I = A - P = 30225.9 - 30000 = 225.9$ **Final answer:** The interest earned is approximately $225.90$.