Noah is going to invest in an account paying an interest rate of \( 7 \% \) compounded continuously. How much would Noah need to invest, to the nearest hundred dollars, for the value of the account to reach \( \$ 12,900 \) in 10 years?

To find out how much Noah needs to invest, we can use the formula for continuous compounding: \[ A = Pe^{rt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the time in years. - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Given: - \( A = 12,900 \) - \( r = 0.07 \) - \( t = 10 \) We need to solve for \( P \). 1. Rearranging the formula gives: \[ P = \frac{A}{e^{rt}} \] 2. Substituting in the known values: \[ P = \frac{12,900}{e^{0.07 \cdot 10}} \] 3. Calculating \( e^{0.7} \): \[ e^{0.7} \approx 2.01375 \] 4. Now substituting back into the equation: \[ P = \frac{12,900}{2.01375} \approx 6,397.77 \] Rounding to the nearest hundred dollars, Noah needs to invest approximately \( \$6,400 \) to reach his goal of \( \$12,900 \) in 10 years at a \( 7\% \) interest rate compounded continuously.