Let's summarize the scenarios for each car and analyze their distance traveled over time.
Car 1:
Distance formula: \( y = t \)
- This indicates that Car 1 travels at a constant speed of 1 mile per hour. As time increases, the distance traveled increases linearly.
Car 2:
Distance values:
- When \( t = 0 \), \( y = 0^2 = 0 \)
- When \( t = 0.2 \), \( y = (0.2)^2 = 0.04 \)
- When \( t = 0.4 \), \( y = (0.4)^2 = 0.16 \)
- When \( t = 0.6 \), \( y = (0.6)^2 = 0.36 \)
- When \( t = 0.8 \), \( y = (0.8)^2 = 0.64 \)
- When \( t = 1 \), \( y = (1)^2 = 1 \)
Analysis:
- Car 2 travels at a speed that increases over time, as the distance is defined by \( y = t^2 \). This shows that the distance traveled is not linear and is a parabolic function, meaning as time goes on, it travels more distance per unit of time.
Car 3:
Distance formula: \( y = 2^t \)
- For various values of \( t \):
- When \( t = 0 \), \( y = 2^0 = 1 \)
- When \( t = 0.1 \), \( y = 2^{0.1} \approx 1.0718 \) (approximately)
- When \( t = 0.2 \), \( y = 2^{0.2} \approx 1.1487 \) (approximately)
- As time progresses, the distance will continue to grow exponentially.
Analysis:
- Car 3 has an exponential mileage. This means the car accelerates rapidly, covering more distance as time moves forward.
Summary of Distances Over Time:
- Car 1 travels linearly (1 mile/hour).
- Car 2 starts slowly and speeds up (quadratic).
- Car 3 travels exponentially, meaning it will quickly cover distances much faster than the others as time increases.
Visual Representation (Optional):
- If you were to graph these distances over time on the same axes, you'd see:
- A straight line for Car 1.
- A parabolic curve for Car 2 that starts flat and rises more steeply.
- An exponential curve for Car 3 that starts at (0,1) and goes upward very steeply.
These graphs would clearly show the differences in each car's speed and how it changes over time. The car that travels the most distance as time progresses is Car 3, while Car 1 would always be the slowest over time.