Problem 68 - Algebra 2

Question

LAW ENFORCEMENT A certain laser device measures vehicle speed to within 3 miles per hour. If a vehicle's actual speed is 65 miles per hour, write and solve an absolute value equation to describe the range of speeds that might register on this device.

Step-by-Step Solution

Answer

The device may register speeds between 62 mph and 68 mph.

Step 1: Identify the Measure of Error

The device measures vehicle speed to within 3 miles per hour. This means that the speed measured could be up to 3 miles per hour more or less than the actual speed.

Step 2: Represent Speed with Absolute Value Equation

To express the possible range of speeds, use the absolute value equation \( |x - 65| = 3 \), where \( x \) is the measured speed and 65 mph is the actual speed.

Step 3: Solve the Absolute Value Equation

The absolute value equation \( |x - 65| = 3 \) means that the difference between the measured speed \( x \) and the actual speed 65 is 3. This gives two equations: 1. \( x - 65 = 3 \) 2. \( x - 65 = -3 \)

Step 4: Solve the First Equation

Add 65 to both sides of the equation \( x - 65 = 3 \) to get: \( x = 68 \).

Step 5: Solve the Second Equation

Add 65 to both sides of the equation \( x - 65 = -3 \) to get: \( x = 62 \).

Step 6: Conclude the Range of Measured Speeds

The solutions from the above equations indicate that the possible range of speeds measured by the laser device is from 62 mph to 68 mph.

Key Concepts

Error Range

In the context of measuring vehicle speed, the term "error range" refers to the possible variation from the actual speed due to the precision of the measuring device. For the laser device mentioned in the exercise, the error range is "+/- 3 miles per hour."
This means that the speed measured might be 3 miles per hour higher or lower than the vehicle's actual speed. To represent this mathematically, we use an absolute value equation. If the actual speed is 65 mph, the measured speed (\( x \)) can vary within this error range, creating an equation of \( |x - 65| = 3\).
Solving this equation helps us identify all possible speeds that might be detected by the device within its error range.

Vehicle Speed Measurement

Measuring a vehicle's speed accurately is crucial for applications ranging from law enforcement to traffic management. Devices like radar and laser speed guns are commonly used by authorities to ensure compliance with speed limits. In our example, the vehicle's actual speed is 65 mph.
The laser device could register a speed between 62 mph and 68 mph due to its error range of 3 mph. This variation is unavoidable due to the device's operational limits and environmental factors, which can affect readings. Understanding the potential error range allows authorities to interpret speed readings accurately and account for possible discrepancies.

• Helps in fair legal enforcement
• Encourages accurate scientific data collection
• Assists in maintaining road safety

Solving Absolute Value Equations

Solving absolute value equations might sound complex, but it's an essential skill in handling real-world data, like our vehicle speed measurements. In the equation \( |x - 65| = 3\), \( x \) represents any potential measured speed. The objective is to find all values of \( x \) that satisfy this equation.
This requires setting up two separate equations:1. \( x - 65 = 3 \)2. \( x - 65 = -3 \)By solving these equations:
- For the first equation, adding 65 to both sides gives us \( x = 68 \).- For the second equation, adding 65 again results in \( x = 62 \).Thus, the speeds that the device might register range from 62 mph to 68 mph. Grasping these steps helps improve problem-solving skills and enables an understanding of how such equations model real-world situations effectively.