Question

PROBLEM SOLVING A study found that a driver's reaction time \(A(x)\) to audio stimuli and his or her reaction time \(V(x)\) to visual stimuli (both in milliseconds) can be modeled by $$ \begin{aligned} &A(x)=0.0051 x^2-0.319 x+15,16 \leq x \leq 70 \\ &V(x)=0.005 x^2-0.23 x+22,16 \leq x \leq 70 \end{aligned} $$ where \(x\) is the age (in years) of the driver. a. Write an inequality that you can use to find the \(x\)-values for which \(A(x)\) is less than \(V(x)\). b. Use a graphing calculator to solve the inequality \(A(x)

Short answer

a. The inequality required for this problem is, \(A(x) < V(x)\). b. By graphing the functions \(A(x)\) and \(V(x)\) within the domain \(16 \leq x \leq 70\) and finding where \(A(x)\) is below \(V(x)\), one can find the solution to the inequality. c. Based on the results, the driver would react more quickly to audio stimuli than visual stimuli in the age range 16 to 70 years.

Step-by-step solution

Create the inequality

From the given, \(A(x) < V(x)\) is the inequality that needs to be solved. This means the reaction time to audio stimuli is less than the reaction time to visual stimuli.

Solve the inequality

To solve the inequality \(A(x)<V(x)\) graphically, the two functions need to be plotted on the same coordinate system within the given domain \(16 \leq x \leq 70\). The solution to the inequality is the set of \(x\)-values for which the graph of \(A(x)\) is below the \(V(x)\) graph.

Determine a reasonable solution

While graphing, you can use the feature of the graphing calculator that finds the intersection points. Each intersection point indicates where the inequality changes from true to false or vice versa. So the range between intersections where \(A(x)<V(x)\) represents a reasonable solution to the problem.

Interpret the result

The solution to the inequality represents when a driver is likely to react faster to audio stimuli than to visual stimuli. So, a driver would likely react more quickly to the siren of an ambulance (audio) than a traffic light changing from green to yellow (visual) in that age range.

Key concepts

Inequality Solving

Solving inequalities is a fundamental skill in algebra, particularly when we're comparing two different functions, like reaction times in the example given. When we set up an inequality, such as\( A(x) < V(x) \), we're stating that we want to find all the values for \( x \) where the driver's reaction time to audio stimuli is quicker than to visual stimuli.

The process involves several critical steps. First, we visualize the two functions on a coordinate plane. We then look for the regions where one function is below the other. This visual aid helps us understand the combined behavior of the functions over a range of values – in this case, the ages of drivers from 16 to 70 years.

The inequality is solved by finding the values of \( x \) that make the inequality true. In practical terms, these \( x \)-values tell us something about the real-life situation we're modeling: when a driver at a certain age will generally have a faster reaction time to sound than to sight.

Graphing Calculator Usage

A graphing calculator is a powerful tool for visualizing and solving mathematical problems, including inequalities. In algebra, these calculators not only plot functions but can also find intersection points, which are key when comparing two functions within a certain domain, such as ages from 16 to 70.

To use a graphing calculator for this exercise, you input the functions \( A(x) \) and \( V(x) \) accordingly. Next, you'll plot both functions on the same graph paying attention to the domain restrictions. Calculators come equipped with zoom features and trace functionalities that allow you to inspect the graph in detail within the specified interval.

The 'Calculate Intersection' feature is particularly useful. It tells us exactly where the two functions meet. This information delineates the boundaries of the solution set for our inequality. What remains is to find the range along the x-axis between these intersection points that satisfy the original inequality \( A(x) < V(x) \), revealing when a driver's auditory reaction time trumps visual.

Mathematical Modeling

Mathematical modeling transforms real-world scenarios into equations and functions that can be analyzed mathematically. In this exercise, we've modeled the reaction times of drivers to audio and visual stimuli as functions of their age. This allows us to use algebra and graphics calculators to compare these reaction times rigorously.

By expressing the driver's reaction times as quadratic functions, we embrace the continuous nature of change in reaction times with age. The models given, \( A(x) = 0.0051 x^2 - 0.319 x + 15 \) and \( V(x) = 0.005 x^2 - 0.23 x + 22 \), encapsulate how each form of stimuli is processed differently by drivers of varying ages. Mathematical modeling gives us a quantitative framework to simulate and predict such behaviors.

Once we've solved the inequalities using a graphing calculator, the models help us infer at which ages a driver might be quicker to respond to audio rather than visual cues. This blending of mathematical theory with real-world application exemplifies the power of mathematical modeling.