Chapter 3.2, Problem 21AYU

(a)

To determine

To find: whether the given relation represents a function or not

Answer to Problem 21AYU

The given relation is not a function.

Explanation of Solution

Given:

Price( $\$$ /Pair),$p$	Demand(Pairs of Jeans Sold per Day),$D$
$20$	$60$
$22$	$57$
$23$	$56$
$23$	$53$
$27$	$52$
$29$	$49$
$30$	$44$

Calculation:

Let us consider the following table

Price( $\$$ /Pair),$p$	Demand(Pairs of Jeans Sold per Day),$D$
$20$	$60$
$22$	$57$
$23$	$56$
$23$	$53$
$27$	$52$
$29$	$49$
$30$	$44$

No, the relation defined by the set of ordered pairs $\left(p,D\right)$ does not define a function.

Because a function cannot have two values at same input, that is, for every $x$ , there corresponds only one $y$ ,although vice-versa is not true. And in this relation, for $p=23,D=56$ and $53$ , hence it is not a function.

Conclusion:

Therefore, the given relation is not a function.

(b)

To determine

To Draw: a scatter diagram of the data.

Answer to Problem 21AYU

Hence,

  

Explanation of Solution

Calculation:

The scatter diagram of the above data is as follows

  

Conclusion:

Hence, the scatter diagram is drawn.

(c)

To determine

To find:the line of best fit

Answer to Problem 21AYU

Therefore, the line of best fit is $D=-1.3355p+86.1974$ .

Explanation of Solution

Calculation:

Using the graphing utility, the line of best fit that models the relation between price and quantity demanded can be found as follows

  $\begin{array}{l}>with(statistics)\\ >X:=Vector\left(\left[20,22,23,23,27,29,30\right],datatype=float\right);\\ X:=\left[\begin{array}{l}20,\\ 22,\\ 23,\\ 23,\\ 27,\\ 29,\\ 30,\end{array}\right]\\ >Y:=Vector\left(\left[60,57,56,53,52,49,44\right],datatype=float\right);\\ Y:=\left[\begin{array}{l}60,\\ 57,\\ 56,\\ 53,\\ 52,\\ 49,\\ 44,\end{array}\right]\\ >Fit\left(a+bt,X,Y,t\right)\\ 86.1973684210525164-1.33552631578946989\end{array}$

Therefore, the line of best fit is $D=-1.3355p+86.1974$ .

Conclusion:

Therefore, the line of best fit is $D=-1.3355p+86.1974$ .

(d)

To determine

To interpret:the slope.

Answer to Problem 21AYU

Hence, the slope is $-1.3355$ .

Explanation of Solution

Calculation:

  $D\left(p\right)=-1.3355p+86.1974$

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 $\text{y-intercept}$ .

Hence, the slope is $-1.3355$ .

Conclusion:

Hence, the slope is $-1.3355$ .

(e)

To determine

To Express:the relationship found in part (c )

Answer to Problem 21AYU

The relationship can be expressed in function notation as $D\left(p\right)=-1.3355p+86.1974$

Explanation of Solution

Calculation:

The relationship can be expressed in function notation as $D\left(p\right)=-1.3355p+86.1974$

Conclusion:

Therefore, the relationship can be expressed in function notation as

  $D\left(p\right)=-1.3355p+86.1974$

(f)

To determine

To find:the domain of the function

Answer to Problem 21AYU

Hence, the domain is $\left[0,64.5\right]$ .

Explanation of Solution

Calculation:

The price cannot be negative; therefore, the domain of the function will be from $0$ to the point where function touches the horizontal axis, that is $D=0$ . So the value of $p$ at $D=0$ should be found

Put $D=0$ in the function obtained as

  $\begin{array}{l}0=-1.3355p+86.1974\\ 1.3355p=86.1974\\ p=64.5432\\ p\approx 64.5\end{array}$

Hence, the domain is $\left[0,64.5\right]$ .

Conclusion:

Hence, the domain is $\left[0,64.5\right]$ .

(g)

To determine

To find: the number of jeans demanded for the price of $\$28$

Answer to Problem 21AYU

Hence, the jeans demanded will be about $49$ .

Explanation of Solution

Calculation:

  $D\left(p\right)=-1.3355p+86.1974$

So if price is $\$28$ a pair, that is, $p=28$ ; jeans demanded can be found as

Put $p=28$ in the function obtained

  $\begin{array}{l}D=-1.3355\left(28\right)+86.1974\\ =-37.394+86.1974\\ =48.8034\\ \approx 49\end{array}$

Hence, the jeans demanded will be about $49$ .

Conclusion:

Hence, the jeans demanded will be about $49$ .