Alang is going to invest in an account paying an interest rate of \( 5.7 \% \) compounded continuously. How much would Alang need to invest, to the nearest dollar, for the value of the account to reach \( \$ 113,000 \) in 5 years?

To determine how much Alang needs to invest today to reach \$113,000 in 5 years with an interest rate of \( 5.7\% \) compounded continuously, we can use the formula for continuous compounding: \[ A = P \cdot e^{rt} \] Where: - \( A \) is the amount of money accumulated after time \( t \), - \( P \) is the principal amount (the initial investment), - \( r \) is the annual interest rate (in decimal), - \( t \) is the time the money is invested for, - \( e \) is the base of the natural logarithm (approximately 2.71828). **Given:** - \( A = \$113,000 \) - \( r = 5.7\% = 0.057 \) - \( t = 5 \) years **Step 1: Rearrange the Formula to Solve for \( P \)** \[ P = \frac{A}{e^{rt}} \] **Step 2: Calculate the Exponent** \[ rt = 0.057 \times 5 = 0.285 \] **Step 3: Compute \( e^{0.285} \)** \[ e^{0.285} \approx 1.32976 \quad (\text{using a Taylor series expansion or a calculator}) \] **Step 4: Calculate \( P \)** \[ P = \frac{113,000}{1.32976} \approx 84,978 \] **Conclusion:** Alang needs to invest **\$84,978** today to reach \$113,000 in 5 years with a continuously compounded interest rate of \( 5.7\% \). **Final Answer:** Alang needs to invest \$84 978 initially.