What is the future value after 12 years if you deposit \( \$ 1,000 \) for your child's education and the interest is guaranteed at \( 1.9 \% \) compounded continuously? (Round your answer to the nearest cent.) \( \$ \)

To find the future value after 12 years with continuous compounding, we can use the formula for continuous compounding: \[ FV = PV \times e^{rt} \] Where: - \( FV \) is the future value - \( PV \) is the present value (initial deposit) - \( r \) is the annual interest rate - \( t \) is the time in years Given: - \( PV = \$1,000 \) - \( r = 1.9\% = 0.019 \) - \( t = 12 \) years Substitute the values into the formula: \[ FV = 1000 \times e^{0.019 \times 12} \] Now, we can calculate the future value using the formula for continuous compounding. Calculate the value by following steps: - step0: Calculate: \(1000e^{0.019\times 12}\) - step1: Multiply the numbers: \(1000e^{0.228}\) - step2: Convert the expressions: \(1000e^{\frac{57}{250}}\) - step3: Rewrite the expression: \(1000\sqrt[250]{e^{57}}\) The future value after 12 years with continuous compounding is approximately $1,256.09.