The table below shows the distance [tex]\( d(t) \)[/tex] in meters that an object travels in [tex]\( t \)[/tex] seconds:

[tex]\[
\begin{tabular}{|c|c|}
\hline
(seconds) & \text{Distance } d(t) \text{ (meters)} \\
\hline
2 & 64 \\
\hline
4 & 256 \\
\hline
6 & 576 \\
\hline
8 & 1024 \\
\hline
\end{tabular}
\][/tex]

What is the average rate of change of [tex]\( d(t) \)[/tex] between 2 seconds and 6 seconds, and what does it represent?

A. [tex]\( 128 \, \text{m/s} \)[/tex]; it represents the average speed of the object between 2 seconds and 6 seconds.

B. [tex]\( 80 \, \text{m/s} \)[/tex]; it represents the average speed of the object between 2 seconds and 6 seconds.

C. [tex]\( 128 \, \text{m/s} \)[/tex]; it represents the average distance traveled by the object between 2 seconds and 6 seconds.

D. [tex]\( 80 \, \text{m/s} \)[/tex]; it represents the average distance traveled by the object between 2 seconds and 6 seconds.



Answer :

To solve this, we need to determine the average rate of change of the distance [tex]\( d(t) \)[/tex] between [tex]\( t = 2 \)[/tex] seconds and [tex]\( t = 6 \)[/tex] seconds. The average rate of change of a function over an interval [tex]\([t1, t2]\)[/tex] is given by the formula:

[tex]\[ \frac{d(t2) - d(t1)}{t2 - t1} \][/tex]

Given the table:

[tex]\[ \begin{tabular}{|c|c|} \hline \text{seconds} & \text{meters} \\ \hline 2 & 64 \\ \hline 4 & 256 \\ \hline 6 & 576 \\ \hline 8 & 1024 \\ \hline \end{tabular} \][/tex]

Here, [tex]\( t1 = 2 \)[/tex] seconds and [tex]\( t2 = 6 \)[/tex] seconds. Correspondingly, [tex]\( d(t1) = 64 \)[/tex] meters and [tex]\( d(t2) = 576 \)[/tex] meters.

Plug these values into the formula:

[tex]\[ \frac{d(t2) - d(t1)}{t2 - t1} = \frac{576 - 64}{6 - 2} \][/tex]

Calculate the numerator:

[tex]\[ 576 - 64 = 512 \][/tex]

Calculate the denominator:

[tex]\[ 6 - 2 = 4 \][/tex]

Now divide the numerator by the denominator:

[tex]\[ \frac{512}{4} = 128 \][/tex]

Hence, the average rate of change of [tex]\( d(t) \)[/tex] between 2 seconds and 6 seconds is [tex]\( 128 \)[/tex] meters per second.

The value [tex]\( 128 \)[/tex] meters per second represents the average speed of the object between [tex]\( t = 2 \)[/tex] seconds and [tex]\( t = 6 \)[/tex] seconds.

Therefore, the correct answer is:
[tex]\[ \boxed{128 \text{ m/s}; \text{ it represents the average speed of the object between 2 seconds and 6 seconds}} \][/tex]