Question

The table below shows the distance d(t) in feet that an object travels in t seconds.

t d(t)
(second) (feet)
1 15
2 60
3 135
4 240

What is the average rate of change of d(t) between 2 seconds and 4 seconds and what does it represent?
A) 50 m/s; it represents the average speed of the object between 2 seconds and 4 seconds
B) 90 m/s; it represents the average speed of the object between 2 seconds and 4 seconds
C) 90 m/s; it represents the average distance traveled by the object between 2 seconds and 4 seconds
D) 50 m/s; it represents the average distance traveled by the object between 2 seconds

Answer 1

B) 90 m/s; it represents the average speed of the object between 2 seconds and 4 seconds

proof

the main formula is v= D / Dt, D=Dfinal - Dinitial
Dt= tfinal - tinitial


so between t=2 and t= 4, V = 240-60/ 4-2= 90m/s

Answer 2

Option B is correct

The average rate of change of d(t) between 2 second and 4 second is; 90 ft/s

and it represents the average speed of the object between 2 seconds and 4 seconds.

Explanation:

Average rate of change of function is defined as the ratio of the difference in the function f(x) as it changes from a to b to the difference between a and b. Then, the average rate of change is denoted as A(x).

`A(x) =(f(b)-f(a))/(b-a)`

As per the given statement, the distance d(t) is in feet and t is the time in second.

To find the average rate of change of d(t) between 2 seconds and 4 seconds.

From the table we have;

at t = 2 , d(2) = 60

and

at t =4 , d(4) = 240.

Then, by the definition of average rate of change ;

`A(t) = (d(4)-d(2))/(4-2)` =
`(240-60)/(4-2) =(180)/(2)`

Simplify:

`A(t) = 90 ft/s`

therefore, the average rate of change of d(t) between 2 second and 4 second is; 90 ft/s and it represents the average speed of the object between 2 seconds and 4 seconds.