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.......
y=2t-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

To compare the speeds of the three cars after 5 hours, let's first analyze the equations and the information given for each car.

**Car 1:**
- The distance is represented by the equation \( y = t \).
- This indicates that the car travels 1 mile for every hour. Thus, after 5 hours, the distance traveled would be \( y = 5 \; \text{miles} \).
- The speed of Car 1 is \( 1 \; \text{mile/hour} \).

**Car 2:**
- The values given suggest a quadratic relationship for distance. Each \( t \) value corresponds to \( t^2 \):
- For \( t = 0 \), \( y = 0 \)
- For \( t = 0.2 \), \( y = 0.04 \)
- For \( t = 0.4 \), \( y = 0.16 \)
- For \( t = 0.6 \), \( y = 0.36 \)
- For \( t = 0.8 \), \( y = 0.64 \)
- For \( t = 1 \), \( y = 1 \)

This suggests the distance equation to be \( y = t^2 \).
- After 5 hours, the distance traveled would be \( y = 5^2 = 25 \; \text{miles} \).
- The speed of Car 2, based on its distance function, is not constant; it varies as speed increases with time. After 5 hours, its speed is \( 10 \; \text{miles/hour} \) since speed can be calculated as the derivative \( \frac{dy}{dt} = 2t \) at \( t = 5 \).

**Car 3:**
- The distance is represented by the equation \( y = 2t - 1 \).
- After 5 hours, the distance traveled would be \( y = 2(5) - 1 = 10 - 1 = 9 \; \text{miles} \).
- The speed of Car 3 is constant at \( 2 \; \text{miles/hour} \) because it is a linear function.

**Comparison After 5 Hours:**
- Car 1: 5 miles at 1 mile/hour.
- Car 2: 25 miles at 10 miles/hour (increasing speed).
- Car 3: 9 miles at 2 miles/hour.

In conclusion, Car 2 is the fastest, traveling 25 miles in 5 hours, followed by Car 3 with 9 miles, and then Car 1 with only 5 miles.