Problem 91 - Contemporary Precalculus

Question

According to data from the National Highway Traffic Safety Administration, the driver fatality rate \(D\) per 1000 licensed drivers every 100 million miles can be approximated by the equation \(D=.0031 x^{2}-.291 x+7.1,\) where \(x\) is the age of the driver. (a) For what ages is the driver fatality rate about 1 death per \(1000 ?\) (b) For what ages is the rate three times greater than in part (a)?

Step-by-Step Solution

Answer

Answer: The driver's age is approximately 17.15 years when the fatality rate is 1 death per 1000 drivers. The fatality rate is three times greater when the driver's age is approximately 20.45 years or 81.46 years.

Step 1: (a) Find the age when fatality rate is 1 death per 1000 drivers

We are given that \(D = 1\). Let's substitute this value into the equation and solve for the age (\(x\)): \(1 = 0.0031x^2 - 0.291x + 7.1\) Now, we need to solve this quadratic equation for \(x\). First, we can subtract 1 from both sides of the equation to set it equal to zero: \(0 = 0.0031x^2 - 0.291x + 6.1\) To find the roots (\(x\) values) of this equation, we can use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Here, \(a = 0.0031, b=-0.291, c=6.1\). Let's plug these values into the quadratic formula to find \(x\): \(x =\frac{-(-0.291) \pm \sqrt{(-0.291)^2 - 4(0.0031)(6.1)}}{2(0.0031)}\) Solving the above equation, we get two possible values for \(x\): \(x \approx 17.15\) or \(x \approx 123\). Since it doesn't make sense to consider a driver age of 123 years, we take only the meaningful solution, which is \(x \approx 17.15\). So, the driver fatality rate is about 1 death per 1000 drivers when the driver's age is approximately 17.15 years.

Step 2: (b) Find the age when the fatality rate is 3 times greater than in part (a)

We need to find the age when the fatality rate is 3 times greater than in part (a). We know that the fatality rate in part (a) is 1 death per 1000 drivers. So, we need to find the age when \(D = 3 \times 1 = 3\). Substitute this value into the equation and solve for \(x\): \(3 = 0.0031x^2 - 0.291x + 7.1\) Subtract 3 from both sides to set it equal to zero: \(0 = 0.0031x^2 - 0.291x + 4.1\) Once again, we will use the quadratic formula to find the roots (\(x\) values): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Here, \(a = 0.0031, b=-0.291, c=4.1\). Let's plug these values into the quadratic formula: \(x = \frac{-(-0.291) \pm \sqrt{(-0.291)^2 - 4(0.0031)(4.1)}}{2(0.0031)}\) Solving the above equation, we get two possible values for \(x\): \(x \approx 20.45\) or \(x \approx 81.46\). As both these age values are plausible, we can conclude that the driver fatality rate is three times greater than in part (a) when the driver's age is approximately 20.45 years or 81.46 years.

Key Concepts

Quadratic Equations

Quadratic equations are a central concept in algebra and precalculus. They are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The driver fatality rate equation presented in the exercise is a particular instance of a quadratic equation.

In the problem, we solve for \(x\) when \(D\), the fatality rate, is given specific values. Solving such equations generally involves rearranging the equation to have zero on one side, as shown in the solution steps. There are several methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula, which is universally applicable and was used in this example.

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is derived from the process of completing the square and provides the roots of any quadratic equation. It's crucial for students to understand the discriminant \(b^2 - 4ac\), which determines the nature of the roots. If the discriminant is positive, there are two real roots; if zero, there is one real root; and if negative, there are no real roots in the set of real numbers.

The solutions found for the driver's age in parts (a) and (b) of the problem are obtained by substituting the constants into the quadratic formula, ensuring we interpret the roots within the context of the problem - discarding non-sensible results like a driver's age of 123 years.

Problem Solving in Precalculus

Problem solving in precalculus involves systematic approaches to tackle mathematical challenges using algebraic and functional concepts. In the context of the driver fatality rate problem, we start by interpreting the given equation and determining what we are solving for - the age of the driver when the fatality rate is at a certain level.

To approach such problems, it is essential to:

• Understand the problem's context and identify what variables represent.
• Set up the equation based on the given or desired conditions.
• Apply appropriate algebraic methods to rearrange and simplify the equation if necessary.
• Utilize problem-solving tools such as the quadratic formula when required.
• Analyze the results within the context of the problem and discard any non-practical solutions.

By breaking down the problem into manageable steps and using the correct tools, such as symbolic manipulation and equations, we make the process of solving complex precalculus problems more accessible and intuitive.

Mathematical Modeling

Mathematical modeling uses mathematical language and equations to represent real-world situations. In our exercise, the driver fatality rate is modeled by a quadratic equation. This equation relates the fatality rate \(D\) to the age of the driver \(x\), allowing us to predict or estimate rates for different ages.

Creating a mathematical model usually involves:

• Identifying key variables and their relationships.
• Using data or theoretical understanding to construct an equation or system of equations.
• Analysing the model to make predictions and draw conclusions.

Mathematical models can range from simple linear models to complex dynamical systems, each providing insights into the mechanism being studied. The provided exercise models the driver fatality rate as a parabolic trend, suggesting that certain ages are associated with higher risks. The equation not only gives us quantitative results but also qualitative understanding, showing that risk is not linearly related to age but instead has a more complicated relationship.

Mathematical modeling is a powerful tool in various fields such as economics, biology, engineering, and social sciences. By translating problems into the language of mathematics, we can leverage the analytical power of mathematical techniques to solve everyday problems, make predictions, and inform decision making.