How much money should you invest now (to the nearest dollar) in an account that earns interest at a rate of \( 7.3 \% \) per year, compounded continuously, in order to have \( \$ 451,000 \) in 9 years? You should invest \( \$ \) Submit answer

To determine how much money you should invest now to have \$451,000 in 9 years with an annual interest rate of 7.3% compounded continuously, we can use the formula for continuous compounding: \[ FV = PV \times e^{rt} \] Where: - \( FV \) is the future value (\$451,000), - \( PV \) is the present value (the amount you need to invest), - \( r \) is the annual interest rate (7.3% or 0.073), - \( t \) is the time in years (9 years), - \( e \) is the base of the natural logarithm (approximately 2.71828). **Step 1: Rearrange the Formula to Solve for \( PV \)** \[ PV = \frac{FV}{e^{rt}} \] **Step 2: Plug in the Known Values** \[ PV = \frac{451,000}{e^{0.073 \times 9}} = \frac{451,000}{e^{0.657}} \] **Step 3: Calculate \( e^{0.657} \)** \[ e^{0.657} \approx 1.929 \] **Step 4: Compute \( PV \)** \[ PV = \frac{451,000}{1.929} \approx 233,800 \] **Conclusion** You should invest **\$233,800** now to have \$451,000 in 9 years with an annual interest rate of 7.3% compounded continuously. \[ \boxed{233800} \]