Let's carefully analyze the given problem and equation to determine the meaning of the variable [tex]\(x\)[/tex] in the context of Sean's jogging routine.
Sean began jogging to live a healthier lifestyle. The problem states:
1. On his first run, he ran one-half mile.
2. He increased his workouts by adding two miles a month to his run.
3. He wrote the equation [tex]\( f(x) = 0.5 + 2x \)[/tex] to model his progress.
The equation provided is [tex]\( f(x) = 0.5 + 2x \)[/tex].
Now, let's break down what each part of the equation represents:
- The term [tex]\( 0.5 \)[/tex] represents the initial distance he ran, which is one-half mile.
- The term [tex]\( 2x \)[/tex] represents the additional miles he adds over time.
To understand what the variable [tex]\( x \)[/tex] stands for, consider the units of the terms and how they contribute to modeling the mileage Sean runs:
- The [tex]\( 0.5 \)[/tex] is a constant, meaning he started with half a mile.
- The [tex]\( 2x \)[/tex] is a variable term that changes each month. The coefficient [tex]\( 2 \)[/tex] indicates that Sean adds 2 miles each month to his run.
Since he adds [tex]\( 2 \)[/tex] miles each month, [tex]\( x \)[/tex] must represent the number of months he has been running. Each month that passes, [tex]\( x \)[/tex] increases by 1, and thus the term [tex]\( 2x \)[/tex] increases accordingly by 2 miles each month.
Let's test it out:
- After 0 months (initially), [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0.5 + 2 \times 0 = 0.5 \text{ miles (initial run)}
\][/tex]
- After 1 month, [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 0.5 + 2 \times 1 = 2.5 \text{ miles (first month run)}
\][/tex]
- After 2 months, [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 0.5 + 2 \times 2 = 4.5 \text{ miles (second month run)}
\][/tex]
In every scenario, [tex]\( f(x) = 0.5 + 2x \)[/tex] correctly models Sean's increasing mileage based on the number of months he has been running.
Therefore, the variable [tex]\( x \)[/tex] represents the number of months he runs.
So the correct answer is:
months he runs.
