Chapter 3.1, Problem 42AYU

Competitive Balance Tax In 2011, major league baseball signed a labor agreement with the players. In this agreement, any team whose payroll exceeded $\$189$ million in 2015 had to pay a competitive balance tax of 50% (for four or more consecutive offenses). The linear function $T\left(p\right)=0.50\left(p-189\right)$ describes the competitive balance tax $T$ of a team whose payroll was $p$ (in millions of dollars).

Source: Major League Baseball

a. What is the implied domain of this linear function?

b. What was the competitive balance tax for the New York Yankees whose 2015 payroll was $\$214.2$ million?

c. Graph the linear function.

d. What was the payroll of a team that paid a competitive balance tax of $\$15.7$ million?

e. Interpret the slope.

To determine

To calculate:

a. What is the implied domain of this linear function?

b. What was the competitive balance tax for the New York Yankees whose 2015 payroll was $\$214.2$ million?

c. Graph the linear function.

d. What was the payroll of a team that paid a competitive balance tax of $\$15.7$ million?

e. Interpret the slope.

Answer to Problem 42AYU

Solution:

a. The domain of the given function is $(189,\infty )$.

b. If the payroll is $214.2$ million, then the player has to pay a competitive balance tax of $\$12.6$ million.

c. The graph is shown below:

d. The payroll will be $\$220.4$ million.

e. If the payroll is increasing then the tax bill also is increasing.

Explanation of Solution

Given:

The given competitive balance tax function is $T(p)=0.50(p-189)$

Calculation:

a. We can see that the variable $x$ in the given linear function can have any value more than 189, therefore, the domain of the given function is $(189,\infty )$.

b. When $p=214.2$, we have

$T(p)=0.50(214.2-189)$

$=12.6$

Therefore, if the payroll is $214.2$ million, then the player has to pay a competitive balance tax of $\$12.6$ million.

c. The graph of the given linear function is

d. If $T(p)=15.7$, then we get

$15.7=0.50(p-189)$

$\Rightarrow p=\frac{15.7}{0.5}+189$

$\Rightarrow p=220.4$

If the team paid a tax of $\$15.7$ million, then the payroll will be $\$220.4$ million.

e. We can see that the slope of the given function is $0.5$ which is a positive and is therefore increasing.

Thus, if the payroll is increasing then the tax bill also is increasing.