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:



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 and describe the functions, let's analyze each car based on the information given.

### Car 1:
The function represented is \( y = t \). This indicates that the distance traveled (y in miles) is equal to the time spent traveling (t in hours). Therefore, after 5 hours:
\[
y = 5 \implies \text{Distance traveled by Car 1} = 5 \, \text{miles}
\]
The speed of Car 1 can be calculated as:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{5}{5} = 1 \, \text{mile per hour}
\]

### Car 2:
The data points for Car 2 are as follows (assuming these are representative of \( t \) values and their corresponding distances traveled):
- At \( t = 0 \): \( y = 0 \)
- At \( t = 0.2 \): \( y = 0.04 \)
- At \( t = 0.4 \): \( y = 0.16 \)
- At \( t = 0.6 \): \( y = 0.36 \)
- At \( t = 0.8 \): \( y = 0.64 \)
- At \( t = 1 \): \( y = 1 \)

Noticing the pattern, we can see that this appears to represent the function:
\[
y = t^2
\]
This means the distance increases with the square of time. After 5 hours:
\[
y = (5)^2 = 25 \, \text{miles}
\]
The speed of Car 2 is therefore:
\[
\text{Speed} = \frac{25}{5} = 5 \, \text{miles per hour}
\]

### Car 3:
Since a description for Car 3 wasn't provided, we generally assume a car would also follow a linear or quadratic path like the first two. However, assuming typical function paths for the context, we could hypothesize that it might resemble constant speed
\( y = kt \) where \( k \) adheres to constant mileages, or a quadratic function like Car 2. Without specific details, we cannot compute the distance or speed of Car 3.

### Summary of Car Speeds After 5 Hours:
- Car 1 travels 5 miles at **1 mile per hour** (linear function \( y = t \)).
- Car 2 travels 25 miles at **5 miles per hour** (quadratic function \( y = t^2 \)).
- Car 3: Unknown without specific details.

If Car 3 can be represented as either being a constant speed or another quadratic function, more information would be needed to conclude accurately.

Based on this data, Car 2 is the fastest, car 1 is the slowest, and conclusions for Car 3 depend on further information.