Q. 42 - Precalculus Enhanced with Graphing Utilities

Question

Luxury Tax In 2002, major league baseball signed a labor agreement with the players. In this agreement, any team whose payroll exceeded \(136.5 million in 2006 had to pay a luxury tax of 40% (for second offenses). The linear function $T\left(p\right)= 0.40\left(p - 136.5\right)$ describes the luxury 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 luxury tax for the New York Yankees whose 2006 payroll was \)171.1 million? 

(c) Graph the linear function. 

(d) What is the payroll of a team that pays a luxury tax of $11.7 million? 

(e) Interpret the slope. 

Step-by-Step Solution

Answer

Part (a) The implied domain of this linear function is $\left\{p|p>136.5\right\}$.

Part (b) The luxury tax was 13.84 million.

Part (c) The graph is 

Part (d) The payroll of a team was $165.75 million.

Part (e) The slope is 0.40. The slope represents the luxury tax increase by $0.40 million for every additional 1 million payroll. 

Part  

Part (a) Step 1 Given information

In 2002, major league baseball signed a labor agreement with the players. that any team whose payroll exceeded $136.5 million in 2006 had to pay a luxury tax of 40% .

The tax T describe as linear function $T\left(p\right)= 0.40\left(p - 136.5\right)$.

Here p is for payroll of the team in millions of dollars. 

Part (a) Step 2. Explanation.

The linear function $T\left(p\right)= 0.40\left(p - 136.5\right)$ describes the luxury tax T of a team. p is for payroll of the team in millions of dollars.

So p is independent variable and T is dependent variable.

As any team exceeded  payroll of $136.5 million had to pay a luxury tax. 

So, the implied domain of this linear function is all the number greater than p. 

Part (b) Step 1. Explanation.

Find the luxury tax, if the payroll of New York Yankees in 2006 was $171.1 million.

We have $p=171.1$

Substitute $p=171.1$  in  $T\left(p\right)=0.40(p - 136.5)$

$$\begin{gathered} T\left(p\right)=0.40(p - 136.5) \\ T\left(171.1\right)=0.40(171.1 - 136.5) \\ T\left(171.1\right)=13.84 \end{gathered}$$

The luxury tax was 13.84 million, if the payroll of New York Yankees in 2006 was $171.1 million. 

Part (c) Step 1. Explanation.

The implied domain of linear function $T\left(p\right)=0.40(p-136.5)$ is $\left[136.5,\infty \right]$

So the graph is 

Part (d) Step 1. Explanation.

Find the payroll of a team that pays a luxury tax of $11.7 million.

We have $T\left(p\right)=11.7$ find p using linear function.

$$\begin{gathered} T\left(p\right)=0.40(p-136.5) \\ 11.7=0.4p-54.6 \\ 66.3=0.4p \\ 165.75=p \end{gathered}$$

The payroll of a team was $165.75 million.

Part (e) Step 1. Explanation.

Interpret the slope.

Compare the function T(p) with standard linear equation $y=mx+b$ where m is slope of the line and b is y-intercept.

The slope represent charge per mile.

In $T\left(p\right)=0.40(p-136.5)$, slope is 0.40.

So, The luxury tax increase by $0.40 million for every additional 1 million payroll.