Problem 90 - College Algebra

Question

The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\) a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

Step-by-Step Solution

Answer

The graph of the function shows a trend of an increasing number of arrests up until a certain age, then it decreases. The age that corresponds to the greatest number of arrests along with that number of arrests can be found using the TRACE or maximum function feature on the graphing utility.

Step 1: Graphing the function

Plot the function \(f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x\) on your graphing utility with the given viewing rectangle [0,70,5] by [0,400,20]. Make sure the x-axis represents the driver's age and the y-axis represents the number of arrests.

Step 2: Describing the trend

Analyze the trend presented by the graph. Look at how the graph changes as x (the driver's age) increases. Pay attention to whether the graph goes up or down as you move from left to right, reaching a peak at some point, and determine what this signifies.

Step 3: Finding the maximum number of arrests

Use ZOOM and TRACE features on the graphing utility to find the age that corresponds to the greatest number of arrests. Take note of the x-value when the y-value is highest on the graph. Additionally, note the highest y-value as it represents the maximum number of arrests per 100,000 drivers for the corresponding age group.

Key Concepts

Graphing Utility

A graphing utility is a tool or software, like a graphing calculator, that allows you to plot mathematical functions and analyze their behavior visually. When dealing with complex functions or when you want to explore how a function changes over a range, a graphing utility becomes indispensable. In the context of the exercise, it's critical to input the function into your graphing utility correctly. This particular function models the relationship between a driver's age and DUI arrests. Make sure your viewing window settings match the parameters given:

• X-axis: From 0 to 70, representing age.
• Y-axis: From 0 to 400, representing arrests per 100,000 drivers.

These settings help visualize the graph appropriately and ensure all important features of the function are visible.

Maximum Feature

The maximum feature in a graphing utility helps identify the highest point on the graph of a function. This is particularly useful in this exercise because you are tasked with finding the age that results in the most arrests per 100,000 drivers. Here's how to use it effectively:

• After inputting your function and setting the viewing window, look for the maximum function or similar feature in your graphing utility.
• Use the TRACE feature to move along the curve. This will show you the values for both axes at each point.
• Pinpoint the highest point, where the y-value reaches its peak. This y-value represents the maximum number of arrests, and the corresponding x-value reveals the driver's age that results in this peak.

Finding and analyzing the maximum helps understand where the function reaches its topmost effective value, which in real-world terms relates to identifying the age group with the highest DUI arrest rate.

Age-Related Trends

Understanding age-related trends in data involves observing how certain variables, in this case, the number of DUI arrests, change as age increases. When you graph the function, you'll notice that as age increases, the number of arrests initially rises, reaches a peak, and then begins to decrease. This trend suggests a particular age group has a higher tendency towards DUI incidents.

• Rising Trend: Indicates that younger drivers may have increasing arrest rates as they age up to a certain point.
• Peak Arrest Age: The age at which the number of arrests hits its maximum, providing a focal point for understanding where interventions could be most effective.
• Decreasing Trend: Suggests that after a certain age, the number of arrests declines, possibly due to more responsible behavior or different lifestyle choices.

These trends can be used to drive policies or preventative measures aimed at reducing DUI incidents among at-risk age groups.

Arrests per 100,000 Drivers

The measure "arrests per 100,000 drivers" is a statistical term that provides a proportional understanding of DUI occurrences within the driving population. It allows for a comparable analysis across different segments of drivers. In this context, it helps understand how widespread DUI incidents are among differing age groups.

• This measure normalizes arrest numbers, providing an equal basis for comparison, regardless of population size, which is critical when comparing age groups.
• Makes trends more visible, showing which age groups have disproportionate numbers, and reflecting the efficacy of certain preventive measures.

Using this statistic, authorities can better understand the impact of DUI incidents related to specific demographics and work towards targeted solutions that address the concerns revealed in this data.