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 \( \$ 1,600 \) 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 \( 8.5 \% \) for the funds, what is the required upfront deposit if we expect a team to win the conference title in 6 years? 8 years? 20 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 cent.)

To find the required upfront deposit for the wager, we need to calculate the present value of the expected donation. The present value formula is given by: \[ PV = \frac{FV}{(1 + r)^n} \] where: - \( PV \) is the present value, - \( FV \) is the future value (the donation amount), - \( r \) is the discount rate, - \( n \) is the number of years. Given: - \( FV = \$1,600 \) (the donation amount), - \( r = 8.5\% \) (the discount rate), - \( n \) varies (6 years, 8 years, 20 years). Let's calculate the present value for each scenario: 1. For 6 years: \[ PV_{6} = \frac{1600}{(1 + 0.085)^6} \] 2. For 8 years: \[ PV_{8} = \frac{1600}{(1 + 0.085)^8} \] 3. For 20 years: \[ PV_{20} = \frac{1600}{(1 + 0.085)^{20}} \] Now, let's calculate the present value for each scenario. Calculate the value by following steps: - step0: Calculate: \(\frac{1600}{\left(1+0.085\right)^{6}}\) - step1: Add the numbers: \(\frac{1600}{1.085^{6}}\) - step2: Convert the expressions: \(\frac{1600}{\left(\frac{217}{200}\right)^{6}}\) - step3: Simplify the expression: \(\frac{1600}{\frac{217^{6}}{200^{6}}}\) - step4: Rewrite the expression: \(\frac{1600\times 200^{6}}{217^{6}}\) Calculate or simplify the expression \( 1600/(1+0.085)^8 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1600}{\left(1+0.085\right)^{8}}\) - step1: Add the numbers: \(\frac{1600}{1.085^{8}}\) - step2: Convert the expressions: \(\frac{1600}{\left(\frac{217}{200}\right)^{8}}\) - step3: Simplify the expression: \(\frac{1600}{\frac{217^{8}}{200^{8}}}\) - step4: Rewrite the expression: \(\frac{1600\times 200^{8}}{217^{8}}\) Calculate or simplify the expression \( 1600/(1+0.085)^{20} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1600}{\left(1+0.085\right)^{20}}\) - step1: Add the numbers: \(\frac{1600}{1.085^{20}}\) - step2: Convert the expressions: \(\frac{1600}{\left(\frac{217}{200}\right)^{20}}\) - step3: Simplify the expression: \(\frac{1600}{\frac{217^{20}}{200^{20}}}\) - step4: Rewrite the expression: \(\frac{1600\times 200^{20}}{217^{20}}\) The required upfront deposit for the wager if we expect a team to win the conference title in: - 6 years: $980.71 - 8 years: $833.07 - 20 years: $312.99 Therefore, the required upfront deposit if we expect a team to win the conference title in 6 years is approximately $980.71.