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

Car 2:

Distance Traveled by Car 2 at Different Time Intervals


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.
▼Click for Long Desc
A graph of the equation y equals 2 to the t power negative1 is shown. It includes the points left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 1 right parenthesis, left parenthesis 2 comma 3 right parenthesis.

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

(2 points)

Answer 1

Car 1: y = t. Linear function — constant speed 1 mi/hr. After 5 hr the car has gone 5 mi and its speed is 1 mph.

Car 2: y = t^2 (table matches t2). Quadratic — speed increases linearly: v = dy/dt = 2t. After 5 hr the car has gone 25 mi and its instantaneous speed is v = 2(5) = 10 mph.

Car 3: y = 2^t − 1 (points (0,0),(1,1),(2,3) match). Exponential — speed grows exponentially: v = dy/dt = (ln 2)·2^t. After 5 hr the car has gone 2^5 −1 = 31 mi and its instantaneous speed is (ln 2)·32 ≈ 22.18 mph.

So after 5 hours Car 3 is fastest, then Car 2, then Car 1.