Question

the standard diameter of a golf ball is 42.67 mm.
a golf ball factory does quality control on the golf
balls it manufactures. golf balls are randomly
measured to ensure the correct size. if the
discrepancy in diameter is more than 0.004 mm,
the production is stopped.
which function could represent this situation?
x f(x) = |x - 42.67|
f(x) = |42.67 - x|
f(x) = |42.67 + x|

Explanation:

Step1: Understand Absolute Value for Discrepancy

The discrepancy between the measured diameter \( x \) and the standard diameter \( 42.67 \) mm is the absolute difference. The absolute value of a difference \( |a - b| \) is equal to \( |b - a| \), so \( |x - 42.67|=|42.67 - x| \). We need a function that represents the absolute difference between the measured diameter \( x \) and the standard \( 42.67 \).

Step2: Analyze Each Function

• For \( f(x)=|x - 42.67| \): This is the absolute difference between \( x \) (measured diameter) and \( 42.67 \) (standard), which represents the discrepancy.
• For \( f(x)=|42.67 - x| \): Since \( |a - b| = |b - a| \), this is equivalent to \( |x - 42.67| \), so it also represents the discrepancy. But wait, the third function \( f(x)=|42.67 + x| \) represents the sum, not the difference, so it's incorrect. Now, between \( |x - 42.67| \) and \( |42.67 - x| \), they are the same. But maybe the first one was marked wrong by mistake? Wait, no—wait, the problem is about discrepancy (difference), so both \( |x - 42.67| \) and \( |42.67 - x| \) are correct in terms of difference, but the third is sum. Wait, maybe the initial mark on \( |x - 42.67| \) was wrong? Wait, no—let's re - check. The discrepancy is \( |\text{measured}-\text{standard}| \) or \( |\text{standard}-\text{measured}| \), both are same. The third function is \( |42.67 + x| \), which is sum, so incorrect. So both \( f(x)=|x - 42.67| \) and \( f(x)=|42.67 - x| \) are correct in terms of difference, but maybe the question has a typo, but among the options, the correct ones are \( f(x)=|x - 42.67| \) (or \( f(x)=|42.67 - x| \)) and the third is wrong. Wait, but the problem is to choose which function represents the situation. The discrepancy is the absolute difference, so either \( |x - 42.67| \) or \( |42.67 - x| \) works, and \( |42.67 + x| \) does not. But since \( |x - 42.67| = |42.67 - x| \), both of the first two are correct in terms of difference, but the third is wrong.

Answer:

The correct functions are \( \boldsymbol{f(x)=|x - 42.67|} \) (or \( \boldsymbol{f(x)=|42.67 - x|} \)) as they represent the absolute discrepancy between the measured diameter \( x \) and the standard diameter \( 42.67 \) mm. The function \( f(x)=|42.67 + x| \) is incorrect as it represents a sum, not a difference. If we have to choose between the first two (since they are equivalent), both \( f(x)=|x - 42.67| \) and \( f(x)=|42.67 - x| \) are correct for representing the discrepancy. But if we consider the options, the first two are correct (and equivalent), and the third is incorrect.