Chapter 1.9, Problem 74PS

Solution

(a)

To Write: The inequality that used to find the $x-$ values for which $A\left(x\right)$ is less than $V\left(x\right)$ .

The inequality is $0.0001x^2-0.089x-7<0$ .

Given information:

Given the expression,

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

  $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Driver's reaction time to audio stimuli $=A\left(x\right)$

Driver's reaction time to video stimuli $=V\left(x\right)$

  $x=$ Drivers age

Calculation:

The objective is to write an inequality that we can find the $x-$ values for which $A\left(x\right)$ is less than $V\left(x\right)$

Since $A\left(x\right)$ is less than $V\left(x\right)$. So, the inequality becomes:

  $A\left(x\right)<V\left(x\right)$

Now, substitute the inequality:

  $\begin{array}{c}0.0051x^2-0.319x+15<0.005x^2-0.23x+22\\ 0.0051-0.319x+15-0.005x^2+0.23x-22<0\\ 0.0001x^2-0.089x-7<0\end{array}$

So, the inequality is: $0.0001x^2-0.089x-7<0$ .

(b)

To Create: The table by finding the solution of the inequality from part (a). And the table should contain $x-$ values from $16$ to $70$ in increments of $6$ .

Given information:

Given the expression,

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

  $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Driver's reaction time to audio stimuli $=A\left(x\right)$

Driver's reaction time to video stimuli $=V\left(x\right)$

  $x=$ Drivers age

Calculation:

Implement the inequality from part $\left(a\right)$

  $y=0.0001x^2-0.089x-7$

$y_{16}$	$-8.3984$
$y_{22}$	$-8.9096$
$y_{28}$	$-9.4136$
$y_{34}$	$-9.9104$
$y_{40}$	$-10.4000$
$y_{46}$	$-10.4000$
$y_{52}$	$-10.8824$
$y_{58}$	$-11.8256$
$y_{64}$	$-12.2864$
$y_{70}$	$-12.7400$

(c)

To Check: The solution that determined in part $\left(b\right)$ by using a graphing calculator to solve the inequality $A\left(x\right)<V\left(x\right)$ graphically. State how the domain $16\le x\le 70$

used to find the reasonable solution.

The inequality is $0.0001x^2-0.089x-7<0$ .

Given information:

Given the expression,

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

  $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Driver's reaction time to audio stimuli $=A\left(x\right)$

Driver's reaction time to video stimuli $=V\left(x\right)$

  $x=$ Drivers age

Graph:

The graph of $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$ and $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$ is shown below:

  

So, the reasonable solution is: $16\le x\le 70.$

(d)

To Interpret: Whether the driver will react more quickly to a traffic light changing

from green to yellow or to the siren of an approaching ambulance based on the results

from parts $\left(b\right)$ and $\left(c\right)$

Yes, the driver will react more quickly to a traffic light changing from green to yellow or to the siren of an approaching ambulance.

Given information:

Given the expression,

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

  $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Driver's reaction time to audio stimuli $=A\left(x\right)$

Driver's reaction time to video stimuli $=V\left(x\right)$

  $x=$ Drivers age

Explanation:

A driver is likely to react more quickly to the siren of an approaching ambulance based on our results from parts $\left(b\right)$ and $\left(c\right)$ . Due to the fact that the audio stimuli reaction time $A\left(x\right)$ is less than the visual stimuli reaction time $V\left(x\right)$ .