Chapter 3.6, Problem 47E

a.

To determine

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

Answer to Problem 47E

The required inequality is $x^2-890x-70000<0$ .

Explanation of Solution

Given information:

The model of the driver’s reaction time $A\left(x\right)$ to audio stimuli: $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

The model of reaction time $V\left(x\right)$ to visual stimuli: $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Age of the driver = x

Calculation:

The given equations are:

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$ and $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Take $A\left(x\right)<V\left(x\right)$ .

  $\begin{array}{c}0.0051x^2-0.319x+15<0.005x^2-0.23x+22\\ 0.0051x^2-0.005x^2-0.319x+0.23x+15-22<0\\ 0.0001x^2-0.089x-7<0\\ 10000\left(0.0001x^2-0.089x-7\right)<\left(0\right)10000\end{array}$

Simplify the above expression further.

  $\begin{array}{c}10000\left(0.0001x^2-0.089x-7\right)<\left(0\right)10000\\ x^2-890x-70000<0\end{array}$

Hence, the required inequality is $x^2-890x-70000<0$ .

b.

To determine

To calculate: The solution of the inequality using a graphing calculator. Also, describe the use of the domain $16\le x\le 70$ to determine a reasonable solution.

Answer to Problem 47E

The required solution of the inequality is $\left(16,70\right)$ .

Explanation of Solution

Given information:

The model of the driver’s reaction time $A\left(x\right)$ to audio stimuli: $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

The model of reaction time $V\left(x\right)$ to visual stimuli: $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Age of the driver = x

Calculation:

The given equations are:

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$ and $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

The graph of inequality is shown below:

  

From the above graph, the interval where the red graph is below the blue graph is the solution of the inequality.

This means the interval $\left(16,70\right)$ is the solution of the inequality.

The given domain helps to obtain the proper solution of the inequality.

Hence, the required solution of the inequality is $\left(16,70\right)$ .

c.

To determine

To determine: Whether the driver would react more quickly to a traffic light changing from green to yellow or a the siren of an approaching ambulance.

Answer to Problem 47E

The driver would react more quickly to a traffic light than to a siren.

Explanation of Solution

Given information:

The model of the driver’s reaction time $A\left(x\right)$ to audio stimuli: $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$

The model of reaction time $V\left(x\right)$ to visual stimuli: $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

Age of the driver = x

Calculation:

The given equations are:

  $A\left(x\right)=0.0051x^2-0.319x+15,16\le x\le 70$ and $V\left(x\right)=0.005x^2-0.23x+22,16\le x\le 70$

The graph of inequality is shown below:

  

From the above graph, the interval where the red graph is below the blue graph is the solution of the inequality.

This means the interval $\left(16,70\right)$ is the solution of the inequality.

The given domain helps to obtain the proper solution of the inequality.

It is seen in the graph that the blue graph (visual stimuli) is above the red graph (audio stimuli). This means that the driver would react more quickly to a traffic light.

Hence, the driver would react more quickly to a traffic light than to a siren.