A professional athlete signs a four-year contract in which the earnings can be modeled by
c = 300,000 + 750,000t,
where t represents the year.
(a) Find the actual value of the athlete's contract.
=7200000
(b) Assuming an annual inflation rate of 4%, what is the present value of the contract? (Round your answer to two decimal places.)
To find the actual value of the athlete's contract, we use the given equation:
c = 300,000 + 750,000t
We plug in the value of t, which represents the fourth year of the contract (since it is a four-year contract):
c = 300,000 + 750,000(4)
c = 300,000 + 3,000,000
c = 3,300,000
Therefore, the actual value of the athlete's contract is $3,300,000.
To calculate the present value of the contract assuming an annual inflation rate of 4%, we need to discount each year's value to its present value. To do this, we can use the present value formula:
P = C / (1 + r)^n
where P is the present value, C is the cash flow, r is the annual inflation rate as a decimal, and n is the number of years.
In this case, the cash flow for each year is given by the equation:
C = 300,000 + 750,000t
So, we need to find the present value for each year and then add them up.
For the first year:
P1 = (300,000 + 750,000(1)) / (1 + 0.04)^1
P1 = 1,050,000 / 1.04
P1 ≈ $1,009,615.38
For the second year:
P2 = (300,000 + 750,000(2)) / (1 + 0.04)^2
P2 = 1,800,000 / 1.0816
P2 ≈ $1,663,734.17
For the third year:
P3 = (300,000 + 750,000(3)) / (1 + 0.04)^3
P3 = 2,550,000 / 1.1259
P3 ≈ $2,268,171.91
For the fourth year:
P4 = (300,000 + 750,000(4)) / (1 + 0.04)^4
P4 = 3,300,000 / 1.1699
P4 ≈ $2,819,021.05
Finally, we add up all the present values:
Present value of the contract = P1 + P2 + P3 + P4
≈ $1,009,615.38 + $1,663,734.17 + $2,268,171.91 + $2,819,021.05
≈ $7,760,542.51
Therefore, the present value of the contract, rounded to two decimal places, is approximately $7,760,542.51.