1. **State the problem:** Rachel wants to find the present value of a bond that will mature to 6000 in 6 years with continuous compounding interest at a rate of 2.5% per year. 2. **Formula used:** The formula for continuous compounding is $$A = Pe^{rt}$$ where: - $A$ is the amount of money accumulated after time $t$, - $P$ is the principal (initial amount), - $r$ is the annual interest rate (decimal), - $t$ is the time in years, - $e$ is Euler's number (approximately 2.71828). 3. **Rearrange the formula to solve for $P$:** $$P = \frac{A}{e^{rt}}$$ 4. **Substitute the known values:** $$P = \frac{6000}{e^{0.025 \times 6}}$$ 5. **Calculate the exponent:** $$0.025 \times 6 = 0.15$$ 6. **Calculate $e^{0.15}$:** $$e^{0.15} \approx 1.16183424$$ 7. **Calculate $P$:** $$P = \frac{6000}{1.16183424}$$ 8. **Show intermediate cancellation:** $$P = \frac{6000}{\cancel{1.16183424}} \approx 5164.87$$ 9. **Final answer:** Rachel should pay approximately **5164.87** now for the bond. This means investing 5164.87 today at 2.5% continuous interest will grow to 6000 in 6 years.