To determine appropriate titles for columns [tex]$A$[/tex] and [tex]$B$[/tex] based on the given data table, we need to analyze the nature of the values in each column.
[tex]\[
\begin{tabular}{|c|c|}
\hline $A$ & $B$ \\
\hline 0 & 0 \\
\hline 4 & 2.5 \\
\hline 8 & 6 \\
\hline 12 & 8 \\
\hline 16 & 10.5 \\
\hline
\end{tabular}
\][/tex]
1. Column A Values: [tex]$0, 4, 8, 12, 16$[/tex]
2. Column B Values: [tex]$0, 2.5, 6, 8, 10.5$[/tex]
### Analysis:
1. Consistency in Column A:
- The values are increasing by 4 units consistently: [tex]\(0 + 4 = 4\)[/tex], [tex]\(4 + 4 = 8\)[/tex], [tex]\(8 + 4 = 12\)[/tex], [tex]\(12 + 4 = 16\)[/tex].
- This pattern suggests that Column A likely represents evenly spaced intervals, a characteristic of time increments.
2. Values in Column B:
- These values are less consistent but generally increasing.
- The pattern suggests these values could be associated with some measure that changes over time, like position or distance.
### Conclusion:
Given:
- The regular intervals in Column A indicate a measure of time.
- The varying values in Column B likely represent a measure that accumulates or changes over time, fitting the definition of position in a motion context.
Thus:
- Column A should be titled "Time".
- Column B should be titled "Position".
### Incorrect Options:
- "Column A should be 'Position,' and Column B should be 'Time'.": This contradicts the regular time intervals observed in Column A.
- "Column A should be 'Velocity,' and Column B should be 'Speed'.": Both velocity and speed measure rates of change, not cumulative values, and thus wouldn't fit the data pattern.
- "Column A should be 'Speed,' and Column B should be 'Velocity'.": Similar to the above, this doesn't fit the cumulative, incremental nature of the data.
Thus, the correct titles are:
- Column A should be "Time," and Column B should be "Position."
