Problem 44 - Algebra 2

Question

LAW ENFORCEMENT Police officers can use the length of skid marks to help determine the speed of a vehicle before the brakes were applied. If the skid marks are on dry concrete, the formula \(\frac{s^{2}}{24}=d\) can be used. In the formula, s represents the speed in miles per hour and \(d\) represents the length of the skid marks in feet. If the length of the skid marks on dry concrete are 50 feet, how fast was the car traveling?

Step-by-Step Solution

Answer

The car was traveling approximately 34.64 miles per hour.

Step 1: Understand the Given Formula

The formula given is \( \frac{s^{2}}{24} = d \), where \( s \) is the speed in miles per hour and \( d \) is the length of the skid marks in feet.

Step 2: Identify the Known Values

We are given that the length of the skid marks \( d \) is 50 feet.

Step 3: Substitute Known Values into the Formula

Substitute \( d = 50 \) into the formula so it becomes \( \frac{s^{2}}{24} = 50 \).

Step 4: Solve for \( s^2 \)

Multiply both sides of the equation by 24 to get \( s^{2} = 50 \times 24 \).

Step 5: Calculate \( s^2 \)

Calculate the right side of the equation: \( 50 \times 24 = 1200 \). So, \( s^{2} = 1200 \).

Step 6: Solve for \( s \)

Take the square root of both sides to find \( s \): \( s = \sqrt{1200} \).

Step 7: Calculate \( s \)

Calculate \( \sqrt{1200} \), which is approximately 34.64. Thus, the speed \( s \) is about 34.64 miles per hour.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, typically taking the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. In this exercise, the formula \( \frac{s^2}{24} = d \) is a rearranged form of a quadratic equation with only a single variable \( s \), representing speed, and a constant value associated with the specific conditions of the skid marks.Quadratic equations frequently feature in algebra due to their capability to model parabolic relations, such as the projectile motion of objects or determining speed and distance. Solving them often involves isolating the squared term, as seen in the step-by-step solution where we found \( s^2 \) by multiplying both sides of the equation by 24. This transforms the equation into a simple multiplication problem, making it easier to solve for the variable. Quadratic equations aren't limited to just academic exercises; they have numerous real-world applications.After isolating \( s^2 \), we solved for \( s \) by taking the square root, a crucial step in finding the original variable value in quadratic problems. The concept of using square roots comes from reversing the squaring operation and is widely used in solving quadratic equations.

Problem Solving

Problem solving in mathematics involves a combination of several smaller tasks to arrive at a solution. In our exercise, the process began with understanding the relationship provided by the formula \(\frac{s^2}{24} = d\), translating word problems into mathematical equations, and solving them by isolating variables.Key steps include:

• Understanding the problem: Grasp the meaning of each term and identify known and unknown values.
• Substitution: Insert known values into the equation to simplify it.
• Isolation of the variable: Manipulate the equation to express the unknown in terms of known quantities.
• Calculation: Apply arithmetic to find the solution, such as multiplying by a constant or taking square roots.

By following these strategic steps, problem solving becomes a structured and manageable process.In this case, applying algebraic manipulation such as multiplying both sides by 24 and taking the square root provides clarity and accuracy, helping to determine practical results. This organized approach is applicable to numerous problems and encourages logical thinking and perseverance in the face of challenging tasks.

Real-World Applications

Quadratic equations and algebraic problem-solving skills have a wide range of applications beyond classroom learning. For instance, they are used in law enforcement to estimate a vehicle's speed based on skid marks.The equation \( \frac{s^2}{24} = d \), used in our exercise, is an example of how mathematics can be applied in traffic accident investigation. Investigators use this formula to reconstruct accidents, determining how fast a car was going before the brakes were engaged based on the length of the skid marks left on dry pavement.Real-world applications demonstrate the practicality of learning algebra. They show how mathematical concepts are not just abstract ideas but tools that can be used for:

• Engineering: Calculating stresses and strains within structures.
• Finance: Analyzing investment options and loan plans.
• Physics: Modeling paths of moving objects.
• Geography: Understanding curves and slopes in landscapes.

These applications emphasize the importance of mastering mathematical problem-solving skills as they translate to real-life scenarios, providing valuable insights and solutions.