1. **State the problem:** We want to find the initial amount $P$ to invest now at an interest rate of 6% per year, compounded continuously, so that the investment grows to $2500 in 5 years. 2. **Formula used:** The formula for continuous compounding is $$A = Pe^{rt}$$ where: - $A$ is the amount after time $t$ - $P$ is the initial principal (what we want to find) - $r$ is the annual interest rate (as a decimal) - $t$ is the time in years - $e$ is Euler's number (approximately 2.71828) 3. **Plug in known values:** $$2500 = P e^{0.06 \times 5}$$ 4. **Simplify the exponent:** $$2500 = P e^{0.3}$$ 5. **Solve for $P$ by dividing both sides by $e^{0.3}$:** $$P = \frac{2500}{e^{0.3}}$$ 6. **Show cancellation step:** $$P = 2500 \times \frac{1}{e^{0.3}} = 2500 \times e^{-0.3}$$ 7. **Calculate $e^{0.3}$ without rounding intermediate steps:** Using a calculator, $e^{0.3} \approx 1.349858807576003$ 8. **Calculate $P$ exactly:** $$P = \frac{2500}{1.349858807576003} \approx 1852.322289$$ 9. **Round to nearest cent:** $$P \approx 1852.32$$ **Final answer:** The initial amount to invest is **1852.32**.