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? $$ \$ \square $$ (Round to the nearest cent.)

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To find the required upfront deposit for the fans to put up their potential donation today, we need to calculate the present value of the expected donation in 6 years, 8 years, and 15 years, given a discount rate of 7.5%.

The formula to calculate the present value (PV) is:
$$ PV = \frac{FV}{(1 + r)^n} $$
where:
- PV = Present Value
- FV = Future Value (the expected donation)
- r = Discount Rate (7.5% in this case)
- n = Number of years

Let's calculate the present value for each scenario:

1. For 6 years:
$$ PV_{6\ years} = \frac{2000}{(1 + 0.075)^6} $$

2. For 8 years:
$$ PV_{8\ years} = \frac{2000}{(1 + 0.075)^8} $$

3. For 15 years:
$$ PV_{15\ years} = \frac{2000}{(1 + 0.075)^{15}} $$

Now, let's calculate the present value for each scenario.
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 the fans to put up their potential donation today is:
- For 6 years: $1295.92
- For 8 years: $1121.40
- For 15 years: $675.93
The required upfront deposits are: - 6 years: \$1,295.92 - 8 years: \$1,121.40 - 15 years: \$675.93

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