Three cars start traveling at the same time. The distance traveled by each is represented in three different ways, with y representing miles traveled and t representing time passed in hours.

Question

Use the images to answer the question.

Car 1:

y=t


Car 2:

0

0.2

0.4

0.6

0.8

1

0

0.04

0.16

0.36

0.64

1

Car 3:

A coordinate plane goes from negative 5 to 5 on the t-axis in intervals of 1 and from negative 5 to 5 on the y-axis in intervals of 1.

Compare the speeds of the 3 cars after 5 hours. In your answer, describe the function represented by each car.

(2 points)

Answer 1

Let's analyze the motion of each car based on the information provided.

Car 1:

The distance function for Car 1 is given by: \[ y = t \] This is a linear function where the distance \( y \) traveled is equal to the time \( t \) in hours. This means that Car 1 travels at a constant speed of 1 mile per hour.

After 5 hours:
Distance: \( y = 5 \) miles.

Car 2:

The distance function for Car 2 appears to be represented in a quadratic form based on the values given. The distances calculated for \( t = 0, 0.2, 0.4, 0.6, 0.8, 1 \) are \( 0, 0.04, 0.16, 0.36, 0.64, 1 \) respectively. This suggests that the function can be represented as: \[ y = t^2 \] This is a quadratic function, indicating that the distance increases with the square of the time. This means that the speed of Car 2 increases over time (accelerating).

After 5 hours:
Distance: \( y = 5^2 = 25 \) miles.

Car 3:

Car 3's distance function is less clearly defined without specific points or values provided in the image description. However, if we assume Car 3 was visualized similarly to Car 2 but with different properties (e.g., some accelerating behavior or higher speeds), we may need to gather more details to define its speed and calculate its distance more accurately.

For instance, if we have a function like \( y = kt^n \) (where \( k \) and \( n \) could define another form of motion), we could understand that it might be more than just linear motion but less than quadratic depending on the power of \( n \). If it's another polynomial function or a different type, we could speculate but can’t conclude without more information.

However, assuming this car had a speed greater than Car 1 but less than Car 2 throughout the period observed, we could proceed with an assumption.

Summary of Speeds After 5 Hours:

• Car 1 (Linear, \[ y = t \]): 5 miles
• Car 2 (Quadratic, \[ y = t^2 \]): 25 miles
• Car 3 (Unspecified): Without more data, we can't provide a precise distance but can conclude that it would likely fall between 5 and 25 miles if it had a speed scaling between linear and quadratic.

Conclusion:

After 5 hours, Car 2 is the fastest, having traveled 25 miles due to its accelerating speed, while Car 1 has traveled 5 miles at a consistent speed of 1 mile per hour. For Car 3, more details are needed to accurately compare its distance.