Problem 40 - A Graphical Approach to College Algebra

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 minimum rate is 17.6 per 100,000 for ages 45-54.

Step 1: Understanding the Function

The function given is a quadratic equation: \(f(x) = 1.8x^2 - 12x + 37.4\). By examining its structure, we know this parabola opens upwards because the coefficient of \(x^2\) is positive (1.8). Our goal is to find the minimum fatality rate and the corresponding age group \(x\).

Step 2: Finding the Vertex

The minimum point of a quadratic function given by \(ax^2 + bx + c\) is located at \(x = -\frac{b}{2a}\). In this case, \(a = 1.8\) and \(b = -12\). Substituting these values gives us \(x = -\frac{-12}{2 \times 1.8} = \frac{12}{3.6}\).

Step 3: Calculating the Vertex

Calculate the value of \(x\) using the formula: \(x = \frac{12}{3.6}\). Performing this division gives \(x = 3.33\). Since \(x\) must be an integer to correspond to a specific age group, we round \(x\) to the nearest whole number, which is \(x = 3\).

Step 4: Determining the Corresponding Age Group

The age groups correspond to integer values of \(x\) with 0 for ages 21-24, 1 for ages 25-34, 2 for ages 35-44, and 3 for ages 45-54. Therefore, \(x = 3\) corresponds to the age group of 45-54.

Step 5: Calculating the Minimum Rate

Substitute \(x = 3\) into the function to find the minimum rate: \(f(3) = 1.8(3)^2 - 12 \times 3 + 37.4\). Compute \(f(3)\): \(f(3) = 1.8 \times 9 - 36 + 37.4 = 16.2 - 36 + 37.4 = 17.6\).

Step 6: Conclusion: Interpretation of Results

The minimum fatality rate occurs for the age group 45-54, and the minimum rate is 17.6 per 100,000 population.

Key Concepts

Parabola

In mathematics, a parabola is a U-shaped curve that can open upwards or downwards. The shape of the parabola is defined by its quadratic equation in the form of \( y = ax^2 + bx + c \). Here, the direction in which the parabola opens depends on the coefficient \(a\).

If \(a\) is positive, the parabola opens upward, like a happy face. This is the case in our function \( f(x) = 1.8x^2 - 12x + 37.4 \), where \( a = 1.8 \). On the other hand, if \(a\) is negative, the parabola opens downward, resembling a sad face.

The parabola's symmetry around its vertex implies

• each side mirrors the other,
• providing a balance to the two halves of the curve.

Understanding the orientation and properties of parabolas is key in solving many real-world problems like the one presented in the exercise.

Vertex of a Parabola

The vertex of a parabola is the most significant point on the curve. It represents the minimum or maximum value of the quadratic function, depending on the direction in which the parabola opens.

For our upward-opening parabola defined by \( f(x) = 1.8x^2 - 12x + 37.4 \), the vertex corresponds to the minimum point. This is where the fatality rate is at its lowest.

To find the vertex, we use the formula \( x = -\frac{b}{2a} \), which gives the vertex's x-coordinate. In our case:

• \( a = 1.8 \),
• \( b = -12 \).

Substituting these values, we get:\[ x = -\frac{-12}{2 \times 1.8} = \frac{12}{3.6} = 3.33 \]

Since the problem is about age groups, \(x\) must be a whole number. We round 3.33 to the nearest integer, which is 3, indicating the vertex lies within the age group of 45-54.

Calculating \(f(3)\) gives the minimum fatality rate at this point, showing how the vertex helps solve this practical situation.

Age Groups

Age groups in this problem help categorize the data into manageable sections. Each group encompasses different ranges of ages, specified by integer values of \(x\):

• \( x = 0 \) for ages 21-24,
• \( x = 1 \) for ages 25-34,
• \( x = 2 \) for ages 35-44,
• \( x = 3 \) for ages 45-54.

Age grouping allows for a simplified summary of demographic trends, particularly when dealing with functions and data like fatality rates.

In our exercise, determining the age group with the minimum rate involves finding the parabola's vertex. Rounding the vertex's x-coordinate to the nearest whole number brings us to \(x = 3\), meaning the age group with the lowest accident fatality rate is 45-54.

This approach offers an efficient way to glean insights from the data as well as real-world applicability in fields like public safety and policy-making.