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? \( \$ 1295.92 \) (Round to the nearest cent.) What is the required upfront deposit if we expect a team to win the conference title in 8 years? \( \$ \) (Round to the nearest cent.)

To calculate the required upfront deposit for the wager, we need to use the present value formula. The present value (PV) of a future amount is given by the formula: \[ PV = \frac{FV}{(1 + r)^n} \] where: - \( PV \) is the present value, - \( FV \) is the future value, - \( r \) is the discount rate, - \( n \) is the number of years. Given that the future value (FV) is $2,000 and the discount rate is 7.5%, we can calculate the present value for different time periods. Let's calculate the present value for each time period: 1. For 6 years: \[ PV = \frac{2000}{(1 + 0.075)^6} \] 2. For 8 years: \[ PV = \frac{2000}{(1 + 0.075)^8} \] 3. For 15 years: \[ PV = \frac{2000}{(1 + 0.075)^{15}} \] Now, let's calculate the present value for each time period. Calculate the value by following steps: - step0: Calculate: \(\frac{2000}{\left(1+0.075\right)^{6}}\) - step1: Add the numbers: \(\frac{2000}{1.075^{6}}\) - step2: Convert the expressions: \(\frac{2000}{\left(\frac{43}{40}\right)^{6}}\) - step3: Simplify the expression: \(\frac{2000}{\frac{43^{6}}{40^{6}}}\) - step4: Rewrite the expression: \(\frac{2000\times 40^{6}}{43^{6}}\) The present value for 6 years is approximately $1295.92. Now, let's calculate the present value for 8 years. Calculate the value by following steps: - step0: Calculate: \(\frac{2000}{\left(1+0.075\right)^{8}}\) - step1: Add the numbers: \(\frac{2000}{1.075^{8}}\) - step2: Convert the expressions: \(\frac{2000}{\left(\frac{43}{40}\right)^{8}}\) - step3: Simplify the expression: \(\frac{2000}{\frac{43^{8}}{40^{8}}}\) - step4: Rewrite the expression: \(\frac{2000\times 40^{8}}{43^{8}}\) The present value for 8 years is approximately $1121.40. Finally, let's calculate the present value for 15 years. Calculate the value by following steps: - step0: Calculate: \(\frac{2000}{\left(1+0.075\right)^{15}}\) - step1: Add the numbers: \(\frac{2000}{1.075^{15}}\) - step2: Convert the expressions: \(\frac{2000}{\left(\frac{43}{40}\right)^{15}}\) - step3: Simplify the expression: \(\frac{2000}{\frac{43^{15}}{40^{15}}}\) - step4: Rewrite the expression: \(\frac{2000\times 40^{15}}{43^{15}}\) The present value for 15 years is approximately $675.93. Therefore, the required upfront deposit for each time period is: - For 6 years: $1295.92 - For 8 years: $1121.40 - For 15 years: $675.93