Question

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.

Use the images to answer the question.

Car 1:

y=t
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:
graph
1,1 2,1/2 3,2.......

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

(2 points)

Answer 1

To compare the speeds of the three cars after 5 hours based on the given functions and graphs, we need to analyze each car's distance function:

### Car 1:
The equation given is \( y = t \). This means that the distance traveled (\( y \)) is equal to the time passed (\( t \)) in hours.

- **Speed**: The speed of Car 1 is constant, at \( 1 \text{ mile/hour} \).
- **After 5 hours**: \( y = 5 \) miles.

### Car 2:
The data points given for Car 2 appear to represent a quadratic function, likely in the form \( y = t^2 \), as evidenced by the points (0,0), (0.2,0.04), (0.4,0.16), (0.6,0.36), (0.8,0.64), and (1,1).

- **Function**: \( y = t^2 \) indicates that the distance traveled increases with the square of time.
- **Speed**: The speed of Car 2 is not constant and increases over time. The speed at any time \( t \) can be found by taking the derivative of the distance function: \( \frac{dy}{dt} = 2t \).
- **After 5 hours**: \( y = 5^2 = 25 \) miles. At that time (t = 5), \( \frac{dy}{dt} = 2(5) = 10 \text{ miles/hour} \).

### Car 3:
Based on the provided graph points (1,1), (2,0.5), (3,2), etc., we can infer that Car 3's distance is a piecewise function or may show non-linear behavior. Notably, the points suggest that the speed is changing:

- **function**: The distance function is not explicitly given but does not appear to represent a linear function due to the unequal intervals of distance traveled. It seems to be a non-linear function where distance can be approximated or modeled but cannot be determined exactly without additional information.
- **Speed**: Like Car 2, the speed of Car 3 changes over time, likely dropping after the initial point, as shown by the observed distance increases.
- **After 5 hours**: We need more details about the function to determine exactly how far Car 3 travels in 5 hours.

### Summary of Comparison:
1. **Car 1**: Travels 5 miles at a constant speed of 1 mile/hour.
2. **Car 2**: Travels 25 miles (increasing speed) with a speed of 10 miles/hour after 5 hours.
3. **Car 3**: Does not provide enough information for a precise distance after 5 hours, though it seems to have a varying speed.

In conclusion, Car 1 is the slowest, Car 2 is the fastest at 10 miles/hour after 5 hours, and Car 3's performance remains uncertain without further specification of its distance function.