In a situation where there is a linear relationship between the time since Joseph started the race and the number of laps he has left to complete, we can express this relationship mathematically.
Let's denote:
- \( t \) as the time in minutes since Joseph started the race.
- \( L(t) \) as the number of laps he has left to complete at time \( t \).
Given the linear relationship, we can write: \[ L(t) = mt + b \] where:
- \( m \) is the slope of the line (this value will be negative since Joseph is completing laps as time progresses),
- \( b \) is the y-intercept of the line, which represents the number of laps Joseph had left when the race started (at \( t = 0 \)).
As time progresses, \( t \) increases, and Joseph completes laps, so the function \( L(t) \) will decrease over time. The slope \( m \) will be a negative value indicating that more laps are being completed as time increases.
For example:
- If Joseph starts with 10 laps remaining (
b = 10) and completes 1 lap every minute (m = -1), we can express this as: \[ L(t) = -1t + 10 \]
After:
- 1 minute: \( L(1) = -1(1) + 10 = 9 \) laps left.
- 3 minutes: \( L(3) = -1(3) + 10 = 7 \) laps left.
This relationship can be used for any set of starting laps and completion rate, allowing us to predict how many laps Joseph has left at any point in the race as long as the linearity holds.