Problem 42 - College Algebra with Corequisite Support

Question

The tolerance for a ball bearing is 0.01 . If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is \(x\) inches, express the tolerance using absolute value notation.

Step-by-Step Solution

Answer

The tolerance condition is expressed as \(|x - 2.0| \leq 0.01\).

Step 1: Understand the Tolerance Specification

The problem gives a tolerance of 0.01 for the ball bearing. Tolerance, in this context, defines how much the actual measured diameter of the ball bearing can deviate from the intended diameter (2.0 inches).

Step 2: Define the Tolerance Range

The tolerance indicates that the measured diameter must fall within 0.01 inches above or below the true diameter. Therefore, the range of acceptable values for the diameter is from 1.99 inches to 2.01 inches.

Step 3: Express Using Absolute Value Notation

To express this tolerance condition in terms of absolute value, we state that the difference between the measured diameter (\(x\)) and the true diameter (2.0) is less than or equal to the tolerance (0.01). The mathematical expression is:\[|x - 2.0| \leq 0.01\]

Key Concepts

Understanding Absolute Value Notation

Absolute value notation provides a mathematical way to express how far a value can differ from a particular number. When dealing with measurements, such as the diameter of a ball bearing, precision is key.
The formula \(|x - a| \leq b\) uses absolute value notation to establish a range of acceptable values around a central value, typically where:

• \(x\) is the measured value.
• \(a\) is the desired or true value.
• \(b\) is the tolerance allowed for deviation.

In our example, \(a\) is 2.0 inches, and the tolerance \(b\) is 0.01. This means we can tolerate variations from the true diameter to ensure the quality of the ball bearing. The inequality \(|x - 2.0| \leq 0.01\) concisely shows that any measurement between 1.99 inches and 2.01 inches is acceptable. Absolute value notation simplifies the representation of such tolerance conditions, making it a highly useful tool in measurement problems.

Understanding Diameter Deviation

When precision is necessary, understanding the concept of diameter deviation is essential. It refers to the permissible change in a measured diameter that deviates from its intended size. This is typically expressed as a range.
In our ball bearing example, the deviation from the true diameter of 2.0 inches is within 0.01 inches, meaning the diameter can vary between:

• 1.99 inches, which is 0.01 inches less than 2.0
• 2.01 inches, which is 0.01 inches more than 2.0

This deviation ensures that the ball bearing fits its intended application without being too tight or too loose.
Properly understanding and managing deviations avoids malfunction or damage to machinery where the parts operate. Thus, knowing how deviations affect measurements and ensuring they remain within specified limits is key to quality control.

Ensuring Measurement Accuracy

Measurement accuracy refers to how closely a measured value aligns with the true value. Little room for error exists when high precision is required, such as in manufacturing.
To optimize measurement accuracy, it’s crucial to minimize errors in readings. These errors can stem from:

• Instrument precision - Ensure the tool used for measurement is capable of fine measurements.
• Human error - Proper technique and careful preparation are necessary.
• Environmental factors - Conditions such as temperature can affect readings.

In this context, achieving a measurement within the allowed tolerance of 0.01 around a central value like 2.0 inches exemplifies high accuracy. Consistently maintaining this level of precision ensures the component functions correctly in its mechanical environment, highlighting the importance of precision instruments and methods.