Mike and Terri estimate that they want to buy a house or \( \$ 214,000.00 \), and they need to make a down payment of \( 16.5 \% \) of the cost of their house. If they nave 30 months to save for the down payment, how nuch do they need to invest into an account earning \( 2.431 \% \) compounded continuously so that they can each their goal? Mike and Terri need to invest Note: Your answer should have a dollar sign and be accurate to two decimal places)

We start by finding the amount needed for the down payment. The house costs \(\$214\,000\) and the down payment is \(16.5\%\) of the cost. Thus, the down payment is \[ A = 0.165 \times 214\,000 = 35\,310. \] Since the interest is compounded continuously at a rate of \(2.431\%\) per year (or \(r = 0.02431\) in decimal) over \(30\) months (which is \(T = 2.5\) years), the amount accumulated from an initial deposit \(P\) is given by \[ A = P e^{rT}. \] We solve for \(P\) by rearranging the formula: \[ P = A e^{-rT} = 35\,310 \, e^{-0.02431 \times 2.5}. \] Calculate the exponent: \[ 0.02431 \times 2.5 = 0.060775. \] So, \[ P = 35\,310 \, e^{-0.060775}. \] Now, compute \(e^{-0.060775}\). (Using a calculator, we find that \(e^{-0.060775} \approx 0.94101\).) Thus, \[ P \approx 35\,310 \times 0.94101 \approx 33\,224.71. \] Therefore, Mike and Terri need to invest \(\$\;33\,224.71\).