Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. As a function of age group \(x,\) the fatality rate (per \(100,000\) population) for males killed in automobile accidents can be approximated by $$f(x)=1.8 x^{2}-12 x+37.4$$ where \(x=0\) represents ages \(21-24, x=1\) represents ages \(25-34, x=2\) represents ages \(35-44,\) and so on. Find the age group at which the accident rate is a minimum, and find the minimum rate. (Source: National Highway Traffic Safety Administration.)

Step-by-Step Solution

Answer

The age group with the minimum rate is 45-54, with a rate of 17.6.

Step 1: Understand the Quadratic Function

The function given is quadratic: \(f(x) = 1.8x^2 - 12x + 37.4\). This kind of function forms a parabola. Since the coefficient of \(x^2\) is positive, the parabola opens upwards, indicating that it has a minimum point.

Step 2: Find the Vertex Formula

The vertex of a parabola \(ax^2 + bx + c\) is at \(x = -\frac{b}{2a}\). This point gives the minimum value for upward-opening parabolas.

Step 3: Calculate the Minimum Age Group (x)

Substitute \(a = 1.8\) and \(b = -12\) into the vertex formula:\[x = -\frac{-12}{2 \times 1.8} = \frac{12}{3.6} = 3.33\approx 3\]The nearest whole number for age groups is \(x = 3\).

Step 4: Calculate the Minimum Fatality Rate

Use \(x = 3\) in the function to find the minimum rate:\[f(3) = 1.8(3)^2 - 12 \times 3 + 37.4 = 1.8 \times 9 - 36 + 37.4 = 16.2 - 36 + 37.4 = 17.6\]Thus, the minimum fatality rate is \(17.6\) per \(100,000\) population.

Step 5: Interpret the Result

The age group \(x = 3\) corresponds to ages \(45-54\). The minimum fatality rate is \(17.6\) deaths per \(100,000\) population.

Key Concepts

Parabolas

Quadratic functions like \(f(x) = 1.8x^2 - 12x + 37.4\) often describe real-world scenarios. They graph as parabolas, which are U-shaped curves. Here's some key points about parabolas:

• The direction of the parabola depends on the sign of the \(x^2\) coefficient. If it's positive, the parabola opens upwards.
• When a parabola opens upwards, it has a lowest point called the vertex. This vertex point indicates the minimum value of the function.
• Parabolas are symmetric around their vertical line of symmetry, which passes through their vertex.
• The vertex is crucial for applications like finding minimum costs, rates, or times in real-life problems.

In this exercise, since the parabola forms an upward-open curve, we focus on finding the vertex to determine the age group with the smallest fatality rate.

Vertex Formula

To find a parabola's vertex, use the vertex formula. For a quadratic equation \( ax^2 + bx + c \), the vertex \( x \)-coordinate is \[-\frac{b}{2a}\]. Here's how to use it:

• Identify constants \( a \) and \( b \). In this case, \( a = 1.8 \) and \( b = -12 \).
• Substitute these into the formula: \( x = -\frac{-12}{2 \times 1.8} \).
• Calculate: \( x = \frac{12}{3.6} \), resulting in 3.33, which rounds to 3.
• This \( x \)-value corresponds to the age group you need.

The vertex point helps in calculating the minimum value of quadratic functions, making it indispensable in scenarios like minimizing accident rates.

Automobile Accident Statistics

Automobile accident statistics track various metrics related to safety and road incidents. Here's why they are crucial:

• They offer insights into high-risk age groups, allowing for targeted safety programs.
• These statistics highlight trends over time, revealing impacts of regulations or innovations.
• They direct resources where they are most needed, like improving road infrastructure or driver education programs.
• Governments and safety organizations rely on these numbers for policy making.

Analyzing such data can help reduce accident rates and improve public safety. For instance, in this problem, we determine which age groups might require more focused safety efforts based on fatality rates calculated by the quadratic model.

Fatality Rate Calculation

Calculating fatality rates helps understand risk levels in various scenarios. Follow these steps to interpret the quadratic model from the exercise:

• Find the minimum rate by substituting the vertex \( x \)-value back into the function \( f(x)\).
• For \( x = 3 \), the function is: \( f(3) = 1.8 \times 3^2 - 12 \times 3 + 37.4 \).
• Calculate step-by-step: \( 1.8 \times 9 = 16.2 \), then subtract \( 36 \) and add \( 37.4 \).
• The computation shows the minimum fatality rate is \( 17.6 \) per \( 100,000\) population.

This result indicates the age group 45-54 is at a lower risk compared to others. Such calculations are vital for crafting effective road safety measures.