Chapter 3.1, Problem 42AYU

(a)

To determine

To find: the implied domain of this linear function.

Answer to Problem 42AYU

The domain of the linear function is $\left\{p|p>128\right\}$ .

Explanation of Solution

Given:

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

Calculation:

The function $T\left(p\right)=0.225\left(p-128\right)$ represents the luxury tax to be paid by a team whose payroll exceeds $128 million. This means that the function applies only to values of $p>128$ .

So, the domain of the linear function is $\left\{p|p>128\right\}$ .

Conclusion:

Therefore, the domain of the linear function is $\left\{p|p>128\right\}$ .

(b)

To determine

To find: the luxury tax for the New York Yankees whose 2006 payroll was $171.1 million.

Answer to Problem 42AYU

The luxury tax for a team whose payroll is $160 million is $7.2 million.

Explanation of Solution

Calculation:

Substitute 160 for p in the given function.

  $T\left(160\right)=0.225\left(160–128\right)$

Evaluate.

  $\begin{array}{l}T\left(160\right)=0.225\left(32\right)\hfill \\ =7.2\hfill \end{array}$

Conclusion:

Therefore, the luxury tax for a team whose payroll is $160 million is $7.2 million.

(c)

To determine

To Graph: the linear function.

Answer to Problem 42AYU

The linear function was drawn.

Explanation of Solution

Calculation:

In order to graph the function, find some points to be plotted on the graph. By using the result from part (b), one point on the graph as $\left(160,7.2\right)$ is obtained. For finding additional points, substitute any value for p which is greater than 129.

Take $p=129$ .

  $\begin{array}{l}T\left(129\right)=0.225\left(129-128\right)\hfill \\ =0.225\hfill \end{array}$

So, one point on the graph will be $\left(129,0.225\right)$

Now, to find one more point, substitute 200 for pin the function.

  $\begin{array}{l}T\left(200\right)=0.225\left(200–128\right)\hfill \\ \begin{array}{l}=0.225\left(72\right)\\ =16.2\end{array}\hfill \end{array}$

Thus, an additional point is $\left(200,16.2\right)$ .

Plot the points on a coordinate plane, and join them using a straight line.

  

Conclusion:

Therefore, the linear function was drawn.

(d)

To determine

To find: the payroll of a team that pays a luxury tax of $11.7 million.

Answer to Problem 42AYU

The payroll of a team that pays a luxury tax of $11.7 million is$180 million.

Explanation of Solution

Calculation:

The objective is to find the value of p when $T\left(p\right)=11.7$ .

  $11.7=0.225\left(p-128\right)$

Divide both the sides by 0.225.

  $\begin{array}{l}\frac{11.7}{0.225}=\frac{0.225\left(p-128\right)}{0.225}\\ 52=p-128\end{array}$

Now, add 128 to both the sides.

  $\begin{array}{l}52+128=p-128+128\hfill \\ 180=p\hfill \end{array}$

Conclusion:

Therefore, the payroll of a team that pays a luxury tax of $11.7 million is$180 million.