What will be the range of the athlete's salary, to the nearest dollar, from the second year to the sixth year he is under contract.

Question

A football player signs a contract with a starting salary of $750,000 per year and an annual increase of 3.5% beginning in the second year. What will be the range of the athlete's salary, to the nearest dollar, from the second year to the sixth year he is under contract.

Answer 1

The domain is ( 2, 3, 4, 5, 6 ) and the values of the range are in grey below:

Answer 2

Questions like this are frequently confusing because of the way percentages work. Keep in mind that the salary will increase by 3.5 percent each year, so this means in the second year, the athlete will earn the base salary of 750,000 as well as an extra 3.5%. To calculate the change in salary, we convert the percentage to a decimal: 3.5% is equal to 0.035 in decimal form. Thus, we can calculate that the athlete earns:

750,000 + 750,000 (0.035) = Salary in Year 2

Using the property of factorization, we can factor the 750,000 from both terms and the formula simplifies to

750,000 (1 + 0.035) = 750,000 (1.035) = Salary in Year 2

The next step is where it gets slightly more confusing; the salary increase of 3.5% in Year 3 is based on the salary earned in Year 2. Following the same logic we used to determine the salary in Year 2 based on the salary in Year 1, we have

750,000 (1.035) * (1.035) = 750,000 (1.035)² = Salary in Year 3

This trend continues, and we can establish the general formula for the athlete's salary as:

750,000 (1.035)^N-1 = Salary in Year N

At this point, we need to ensure that we answer the question that is actually being asked: what is the range of the salary from the second year to the sixth year? Well, in the second year, the athlete earns (to the nearest dollar) $776,250 while in the sixth year, the athlete earns $890,765. The range is the interval of these two numbers; in particular, the range is from $776,250 to $890,765.

I hope this helps explain how to do the problem! Let me know if you find anything confusing and I'll try to clarify the explanation.

Answer 3

S(2) = 750000(1.035)^2-1

S(6) = 760000(1/035)^6-1