The standard diameter of a golf ball is [tex]$42.67 \, \text{mm}[tex]$[/tex]. A golf ball factory performs quality control on the balls it manufactures by randomly measuring them to ensure the correct size. One day, an inspector decides to stop production if the discrepancy in diameter is more than [tex]$[/tex]0.002 \, \text{mm}$[/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 :

Sure! To identify which function could represent the discrepancy in the diameter of the golf balls, let's break down the problem step-by-step.

### Problem Restatement
1. The standard diameter of a golf ball is \( 42.67 \, \text{mm} \).
2. We need a function that calculates the discrepancy of measured diameters from this standard.
3. The discrepancy in diameter should not exceed \( 0.002 \, \text{mm} \). If it does, production is stopped.

The goal is to find a function \( f(x) \) that represents how far a measured diameter \( x \) is from the standard diameter \( 42.67 \, \text{mm} \).

### Step-by-Step Solution:

1. Understand the Discrepancy Calculation:
- Discrepancy - the absolute difference between the measured diameter \( x \) and the standard \( 42.67 \, \text{mm} \).
- This can be calculated as: \( \lvert 42.67 - x \rvert \).

2. Formulate the Problem in Terms of the Discrepancy Function:
- We need to find \( f(x) \) such that \( f(x) = \lvert 42.67 - x \rvert \).

3. Break Down Each Given Function:
- \( f(x) = x - \lvert 42.67 \rvert \):
- This expression is not correct as it simplifies to \( x - 42.67 \), which is not an absolute difference.
- \( f(x) = \lvert x \rvert - 42.67 \):
- This expression also is not correct because it subtracts the standard diameter from the absolute value of \( x \).
- \( f(x) = \lvert 42.67 - x \rvert \):
- This is exactly the discrepancy calculation required. It captures the absolute difference between the measured diameter and the standard.
- \( f(x) = 42.67 - \lvert x \rvert \):
- This expression subtracts the absolute value of \( x \) from the standard diameter, which is not correct as the discrepancy function.

### Conclusion
The correct function that represents the discrepancy in the diameter is:
[tex]\[ f(x) = \lvert 42.67 - x \rvert \][/tex]

Thus, the appropriate choice is:
[tex]\[ f(x) = \lvert 42.67 - x \rvert \][/tex]

Hence, we see that the inspector should use this function to ensure the diameters remain within the allowable discrepancy limit of [tex]\( 0.002 \, \text{mm} \)[/tex].