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]
