The standard diameter of a golf ball is [tex]$42.67 \text{ mm}$[/tex]. A golf ball factory does quality control on the balls it manufactures. Golf balls are randomly measured to ensure the correct size. One day, an inspector decides to stop production if the discrepancy in diameter is more than [tex]$0.002 \text{ mm}$[/tex]. Which function could represent this situation?

A. [tex]f(x) = x - |42.67|[/tex]
B. [tex]f(x) = |x| - 42.67[/tex]
C. [tex]f(x) = |42.67 - x|[/tex]
D. [tex]f(x) = 42.67 - |x|[/tex]



Answer :

To determine which function represents the discrepancy in the diameter of the golf ball as described, let's break down the problem step-by-step:

1. Understanding the Problem:

- The standard diameter of the golf ball is [tex]\(42.67 \, \text{mm}\)[/tex].
- The discrepancy we're concerned about is the difference between the measured diameter [tex]\(x\)[/tex] of a golf ball and the standard diameter [tex]\(42.67 \, \text{mm}\)[/tex].
- We are looking for a function that captures this difference, expressed as the absolute value of the difference since discrepancies can be both positive and negative.

2. Formulating the Discrepancy:

- The key element here is the absolute difference between [tex]\(x\)[/tex] and [tex]\(42.67\)[/tex].
- This absolute difference can be mathematically represented as [tex]\(|42.67 - x|\)[/tex].

3. Evaluating Given Functions:

- [tex]\(f(x) = x - |42.67|\)[/tex]:

This implies subtracting the absolute value of [tex]\(42.67\)[/tex] (which remains [tex]\(42.67\)[/tex]) from [tex]\(x\)[/tex]. This does not account for the absolute difference but rather a simple subtraction operation.

- [tex]\(f(x) = |x| - 42.67\)[/tex]:

This takes the absolute value of the measurement [tex]\(x\)[/tex] and subtracts [tex]\(42.67\)[/tex]. This function does not correctly represent the absolute difference between [tex]\(x\)[/tex] and [tex]\(42.67\)[/tex].

- [tex]\(f(x) = |42.67 - x|\)[/tex]:

This is the absolute value of the difference between [tex]\(42.67\)[/tex] and [tex]\(x\)[/tex], which perfectly captures the discrepancy between the measured diameter and the standard diameter.

- [tex]\(f(x) = 42.67 - |x|\)[/tex]:

This incorrectly takes the absolute value of [tex]\(x\)[/tex] first and then subtracts it from [tex]\(42.67\)[/tex], which does not accurately reflect the requirement for the absolute difference.

4. Conclusion:

Based on the requirement to accurately capture the discrepancy between the measured diameter and the standard diameter of [tex]\(42.67 \, \text{mm}\)[/tex]:

The appropriate function that represents this situation is:

[tex]\[ f(x) = |42.67 - x| \][/tex]

This function properly measures the absolute discrepancy in the diameter of the golf balls, ensuring quality control can determine if the difference exceeds the allowable [tex]\(0.002 \, \text{mm}\)[/tex].

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