Chapter 3.1, Problem 51AYU

(a)

To determine

To draw: plot the ordered pairs $\left(p,q\right)$ in a Cartesian plane.

Answer to Problem 51AYU

The Cartesian plane is plotted.

Explanation of Solution

Given:

Let us consider the following table

Price, p (in dollars)	Quantity Demanded, q
150	100
200	80
250	60
300	40

Calculation:

The ordered pairs $\left(p,q\right)$ in Cartesian Plain is as follows

  

Conclusion:

Therefore, the Cartesian plane is plotted.

(b)

To determine

To show: that quantity demanded q is a linear function of the price p .

Explanation of Solution

Calculation:

if average rate of change is constant then function is linear.

Hence, find average rate of change of the function at different points as

Points	Average Rate of Change,   $\frac{\Delta q}{\Delta p}$
$\left(150,100\right)\text{ and }\left(200,80\right)$	$\frac{80-100}{200-150}=\frac{-20}{50}=-\frac{2}{5}$
$\left(200,80\right)\text{ and }\left(250,60\right)$	$\frac{60-80}{250-200}=\frac{-20}{50}=-\frac{2}{5}$
$\left(250,60\right)\text{ and }\left(300,40\right)$	$\frac{40-60}{300-250}=\frac{-20}{50}=-\frac{2}{5}$

It is clear that the average rate of function is constant; hence quantity demanded q is a linear function of the price p .

Conclusion:

Therefore, the single filer's tax bill $=\$2582.50$

(c)

To determine

the linear function that describes the relation between p and q .

Answer to Problem 51AYU

The linear function that describes the relation between p andq is $q\left(p\right)=-0.4p+160$ .

Explanation of Solution

Calculation:

To determine the linear function describing the relation between p and q , let us consider any two points, say, $\left(150,100\right)$ and $\left(200,80\right)$

the equation of the line passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is $\left(y-y_1\right)=\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)$

Similarly, equation of line passing through $\left(150,100\right)$ and $\left(200,80\right)$ is

  $\begin{array}{l}\left(y-100\right)=\left(\frac{80-100}{200-150}\right)\left(x-150\right)\\ y-100=\frac{-20}{50}\left(x-150\right)\\ y-100=-\frac{2}{5}x+60\\ y=-\frac{2}{5}x+160\end{array}$

Where $y=q$ and $x=p,$ therefore

  $q=-\frac{2}{5}p+160$

Hence, the linear function that describes the relation between p andq is $q\left(p\right)=-0.4p+160$ .

Conclusion:

Therefore, The linear function that describes the relation between p andq is $q\left(p\right)=-0.4p+160$ .

(d)

To determine

To find: the implied domain of the linear function.

Answer to Problem 51AYU

The implied domain of the linear function is $\left[0,400\right]$ .

Explanation of Solution

Calculation:

The graph of the linear function cannot be negative, therefore, the graph will terminate there where it becomes zero.

  $\begin{array}{l}\begin{array}{l}q=0\\ -0.4p+160=0\end{array}\hfill \\ 0.4p=160\hfill \\ p=400\hfill \end{array}$

Hence, the implied domain of the linear function is $\left[0,400\right]$ .

Conclusion:

Therefore, the implied domain of the linear function is $\left[0,400\right]$ .

(e)

To determine

To graph: the linear function in the Cartesian plane drawn in part (a)

Answer to Problem 51AYU

The graph is drawn.

Explanation of Solution

Calculation:

The graph of the linear function in the Cartesian plane drawn in part (a) is as follows

  

Conclusion:

Therefore, the graph is drawn.

(f)

To determine

To interpret: the slope

Answer to Problem 51AYU

The slope of the function $q\left(p\right)=-0.4p+160$ is $-0.4$ .

Explanation of Solution

Calculation:

  $q\left(p\right)=-0.4p+160$ .

The linear function is of the form $f\left(x\right)=mx+b$ , where m is the slope of the graph of the function which is straight line and b is the y -intercept.

Hence, slope of the function $q\left(p\right)=-0.4p+160$ is $-0.4$ .

Conclusion:

Therefore, the slope of the function $q\left(p\right)=-0.4p+160$ is $-0.4$ .

(g)

To determine

To interpret: the values of the intercepts

Answer to Problem 51AYU

Thep -intercept is 400.

Explanation of Solution

Calculation:

The q -intercept is 160. And p -intercept can be found by putting $q=0$ in the function as

  $\begin{array}{l}\begin{array}{l}0=-0.4p+160\\ 0.4p=160\end{array}\hfill \\ p=400\hfill \end{array}$

Hence, p -intercept is 400.

Conclusion:

Therefore, the p -intercept is 400.