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.
(b) Assuming an annual inflation rate of 4%, what is the present value of the contract? (Round your answer to two decimal places.)
(a) ∫[0,4] 300000+750000t dt = 7,200,000
Now you can use that to find the present value
I COULDNT FIND IT CAN YOU PLEASE HELP ME HERE ?
(a) To find the actual value of the athlete's contract, we need to determine the earnings for each year and sum them up.
The given formula for earnings is c = 300,000 + 750,000t, where t represents the year.
Since the contract is for four years, we need to evaluate the formula for each year from t = 1 to t = 4:
For t = 1:
c1 = 300,000 + 750,000(1) = 300,000 + 750,000 = 1,050,000
For t = 2:
c2 = 300,000 + 750,000(2) = 300,000 + 1,500,000 = 1,800,000
For t = 3:
c3 = 300,000 + 750,000(3) = 300,000 + 2,250,000 = 2,550,000
For t = 4:
c4 = 300,000 + 750,000(4) = 300,000 + 3,000,000 = 3,300,000
Now, we can sum up the earnings for each year:
c = c1 + c2 + c3 + c4
c = 1,050,000 + 1,800,000 + 2,550,000 + 3,300,000
c = 8,700,000
Therefore, the actual value of the athlete's contract is $8,700,000.
(b) To find the present value of the contract considering an annual inflation rate of 4%, we will discount the earnings for each year to their present values and then sum them up.
The present value (PV) of an amount A after t years with an annual inflation rate of r% can be calculated using the formula:
PV = A / (1 + r/100)^t
Using this formula, we can find the present value for each year:
For t = 1:
PV1 = 1,050,000 / (1 + 4/100)^1 = 1,050,000 / 1.04 ≈ 1,009,615.38
For t = 2:
PV2 = 1,800,000 / (1 + 4/100)^2 = 1,800,000 / 1.0816 ≈ 1,664,692.58
For t = 3:
PV3 = 2,550,000 / (1 + 4/100)^3 = 2,550,000 / 1.125 + 1 ≈ 2,267,716.82
For t = 4:
PV4 = 3,300,000 / (1 + 4/100)^4 = 3,300,000 / 1.1696 ≈ 2,822,580.65
Now, we can sum up the present values for each year:
PV = PV1 + PV2 + PV3 + PV4
PV = 1,009,615.38 + 1,664,692.58 + 2,267,716.82 + 2,822,580.65
PV ≈ 7,764,605.43
Therefore, the present value of the contract considering an annual inflation rate of 4% is approximately $7,764,605.43.
To find the actual value of the athlete's contract, we can substitute the value of t into the given equation c = 300,000 + 750,000t.
(a)
For a four-year contract, the value of t will be 4.
Substituting t = 4 into the equation, we have:
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.
(b)
To find the present value of the contract taking inflation into account, we need to discount the future earnings by the inflation rate.
The present value (PV) of an amount A after t years at an annual inflation rate of r can be calculated using the formula:
PV = A / (1 + r)^t
In this case, the earnings A is $3,300,000, the inflation rate r is 4% (0.04), and the time t is 4 years.
Using the formula, we can calculate the present value:
PV = 3,300,000 / (1 + 0.04)^4
PV = 3,300,000 / (1.04)^4
PV = 3,300,000 / 1.16993
PV ≈ 2,816,409.29
Therefore, the present value of the contract, considering an annual inflation rate of 4%, is approximately $2,816,409.29 (rounded to two decimal places).