Let's begin by examining the equation given in the problem: [tex]\( f(x) = 0.5 + 2x \)[/tex].
This equation helps model Sean's running progress, where:
- [tex]\( f(x) \)[/tex] is the total distance in miles that Sean runs at a given time.
- [tex]\( 0.5 \)[/tex] represents the number of miles Sean ran on his first run.
- [tex]\( 2x \)[/tex] represents the additional miles Sean runs after running for [tex]\( x \)[/tex] months, given that he adds 2 miles per month to his running routine.
To determine what the variable [tex]\( x \)[/tex] represents, let's consider each part of the description:
1. Miles he runs: This cannot be the correct interpretation of [tex]\( x \)[/tex] because in the equation [tex]\( x \)[/tex] is multiplied by 2, indicating that it is not directly describing the number of miles but something else that results in an increase in miles.
2. Months he runs: This is a possible option because it fits the context where he is increasing his distance by 2 miles each month.
3. Miles he ran the first day: This option doesn't fit because the miles he ran the first day are fixed and represented by 0.5, not [tex]\( x \)[/tex].
4. Calories he burns: This interpretation doesn't make sense in the context of the equation given, since the equation specifically models the distance run, not the calories burned.
Given that Sean increases his running distance by 2 miles each month, the variable [tex]\( x \)[/tex] logically represents the number of months he has been running. As he continues to run for more months, the value of [tex]\( x \)[/tex] increases, thus increasing the total distance he runs.
Therefore, the correct interpretation of the variable [tex]\( x \)[/tex] is that it represents the number of months he runs.
