Present value. Two rival football fans have made the following wager: if one fan's college football team wins the conference title outright, the other fan will donate \( \$ 2,000 \) to the winning school. Both schools have had relatively unsuccessful teams, but are improving each season. If the two fans must put up their potential donation today and the discount rate is \( 7.5 \% \) for the funds, what is the required upfront deposit if we expect a team to win the conference title in 6 years? 8 years? 15 years? What is the required upfront deposit if we expect a team to win the conference title in 6 years? \( \$ \square \) (Round to the nearest ceit.)

To determine the required upfront deposit, we can use the present value formula: \[ PV = \frac{FV}{(1 + r)^n} \] Where: - \( PV \) is the present value (upfront deposit), - \( FV \) is the future value of the donation (\$2,000), - \( r \) is the discount rate (7.5% or 0.075), and - \( n \) is the number of years until the payment is made. Calculating for 6 years: \[ PV = \frac{2000}{(1 + 0.075)^6} \approx \frac{2000}{1.484812} \approx 1346.43 \] Thus, the required upfront deposit if we expect a team to win the conference title in 6 years is approximately **\$1346.43** (rounded to the nearest cent). Calculating for 8 years: \[ PV = \frac{2000}{(1 + 0.075)^8} \approx \frac{2000}{1.850931} \approx 1080.33 \] Required upfront deposit for 8 years is approximately **\$1080.33**. Calculating for 15 years: \[ PV = \frac{2000}{(1 + 0.075)^{15}} \approx \frac{2000}{2.849881} \approx 700.18 \] Finally, the required upfront deposit if we expect a team to win the conference title in 15 years is approximately **\$700.18**.