1. **State the problem:** We want to find the initial investment amount $P$ that will grow to $2500$ in 5 years with a continuous compounding interest rate of 6% per year. 2. **Formula used:** The continuous compound interest formula is: $$A = Pe^{rt}$$ where: - $A$ is the amount after time $t$ - $P$ is the principal (initial investment) - $r$ 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:** $$2500 = P e^{0.06 \times 5}$$ $$2500 = P e^{0.3}$$ 4. **Solve for $P$:** $$P = \frac{2500}{e^{0.3}}$$ 5. **Intermediate step showing cancellation:** $$P = \frac{2500}{\cancel{e^{0.3}}} \times \cancel{e^{-0.3}} = 2500 e^{-0.3}$$ 6. **Calculate $e^{0.3}$:** $$e^{0.3} \approx 1.349858807576003$$ 7. **Calculate $P$:** $$P = \frac{2500}{1.349858807576003} \approx 1852.0455$$ 8. **Round to nearest cent:** $$P \approx 1852.05$$ **Final answer:** You should invest approximately **1852.05** now to have 2500 in 5 years with continuous compounding at 6% per year.