The National Science Foundation (NSF) offers grants to students who are interested in pursuing a college education. The student must be enrolled full time ( 12 credit hours or more), and hold a grade point average of 3.8 . The NSF will pay \( \$ 9,500 \) per year towards tuition for 4 years. The initial amount is placed into an account paying \( 7.8 \% \) compounded annually, and the tuition is drawn from the account until the remaining balance is zero dollars. Determine the initial amount invested by the NSF. Round your answer to the nearest dollar. \( \begin{array}{ll}\text { a. } \$ 30,000 & \text { c. } \$ 35,036 \\ \text { b. } \$ 31,606 & \text { d. } \$ 38,000\end{array} \)

To determine the initial amount invested by the NSF, we need to calculate the present value of the tuition payments that will be made over the 4 years. The NSF will pay \( \$ 9,500 \) per year for 4 years, and the account earns \( 7.8\% \) interest compounded annually. The formula for the present value \( PV \) of an annuity (which is what we have here, since the NSF is making equal payments each year) is given by: \[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] where: - \( P \) is the annual payment (\$9,500), - \( r \) is the annual interest rate (7.8% or 0.078), - \( n \) is the number of years (4). Now, substituting the values into the formula: \[ PV = 9500 \times \left( \frac{1 - (1 + 0.078)^{-4}}{0.078} \right) \] Calculating \( (1 + 0.078)^{-4} \): \[ (1 + 0.078)^{-4} = (1.078)^{-4} \approx 0.7548 \] Now substituting this back into the present value formula: \[ PV = 9500 \times \left( \frac{1 - 0.7548}{0.078} \right) \] Calculating \( 1 - 0.7548 \): \[ 1 - 0.7548 = 0.2452 \] Now substituting this value: \[ PV = 9500 \times \left( \frac{0.2452}{0.078} \right) \] Calculating \( \frac{0.2452}{0.078} \): \[ \frac{0.2452}{0.078} \approx 3.1423 \] Now substituting this back into the present value calculation: \[ PV = 9500 \times 3.1423 \approx 29851.85 \] Rounding to the nearest dollar, we find: \[ PV \approx 29852 \] Now, we need to check which of the provided options is closest to this value. The options are: - a. \$30,000 - b. \$31,606 - c. \$35,036 - d. \$38,000 The closest option to \$29,852 is: **a. \$30,000**.