Answer :
First, let's analyze the chart provided:
[tex]\[ \begin{array}{|c|c|} \hline A & B \\ \hline 0 & 0 \\ \hline 4 & 2.5 \\ \hline 8 & 6 \\ \hline 12 & 8 \\ \hline 16 & 10.5 \\ \hline \end{array} \][/tex]
We will start by determining the relationship between columns [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
#### Step 1: Determine the Relationship
To find this, we can calculate the slope between any two points. The slope [tex]\((m)\)[/tex] is given by:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's use the points [tex]\( (0, 0) \)[/tex] and [tex]\( (16, 10.5) \)[/tex].
[tex]\[ m = \frac{10.5 - 0}{16 - 0} = \frac{10.5}{16} = 0.65625 \][/tex]
The slope we obtained is 0.65625. This slope represents the rate of change of [tex]\( B \)[/tex] with respect to [tex]\( A \)[/tex].
#### Step 2: Determine the Y-intercept
Since one of the recorded points is [tex]\( (0, 0) \)[/tex], we can see that the y-intercept [tex]\( (b) \)[/tex] is:
[tex]\[ b = 0 \][/tex]
Thus, the equation of the line is:
[tex]\[ y = 0.65625x + 0 \][/tex]
#### Step 3: Infer the Context Based on Slope and Graph
Given the slope and y-intercept, the relationship suggests a linear change, which is typical in physics for phenomena such as:
- Position as a function of time
- Velocity as a function of time under constant acceleration
#### Step 4: Analyze Possible Titles
The plausible titles for columns need to match the context of a linear relationship where the slope suggests a steady increase over time.
1. Column A should be "Time," and Column B should be "Position.": Time versus position graphs usually exhibit a linear relationship, especially under constant velocity.
2. Column A should be "Position," and Column B should be "Time.": This would imply time changes with position, less intuitive for this data.
3. Column A should be "Velocity," and Column B should be "Speed.": This doesn't align well with the linear relationship and absolute values given.
4. Column A should be "Speed," and Column B should be "Velocity": Similar to above, this doesn't match the provided data context.
Thus, the best fit titles are:
Column A should be "Time," and Column B should be "Position."
This matches well with the slope and intercept obtained, indicating a typical position change over time relationship.
[tex]\[ \begin{array}{|c|c|} \hline A & B \\ \hline 0 & 0 \\ \hline 4 & 2.5 \\ \hline 8 & 6 \\ \hline 12 & 8 \\ \hline 16 & 10.5 \\ \hline \end{array} \][/tex]
We will start by determining the relationship between columns [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
#### Step 1: Determine the Relationship
To find this, we can calculate the slope between any two points. The slope [tex]\((m)\)[/tex] is given by:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's use the points [tex]\( (0, 0) \)[/tex] and [tex]\( (16, 10.5) \)[/tex].
[tex]\[ m = \frac{10.5 - 0}{16 - 0} = \frac{10.5}{16} = 0.65625 \][/tex]
The slope we obtained is 0.65625. This slope represents the rate of change of [tex]\( B \)[/tex] with respect to [tex]\( A \)[/tex].
#### Step 2: Determine the Y-intercept
Since one of the recorded points is [tex]\( (0, 0) \)[/tex], we can see that the y-intercept [tex]\( (b) \)[/tex] is:
[tex]\[ b = 0 \][/tex]
Thus, the equation of the line is:
[tex]\[ y = 0.65625x + 0 \][/tex]
#### Step 3: Infer the Context Based on Slope and Graph
Given the slope and y-intercept, the relationship suggests a linear change, which is typical in physics for phenomena such as:
- Position as a function of time
- Velocity as a function of time under constant acceleration
#### Step 4: Analyze Possible Titles
The plausible titles for columns need to match the context of a linear relationship where the slope suggests a steady increase over time.
1. Column A should be "Time," and Column B should be "Position.": Time versus position graphs usually exhibit a linear relationship, especially under constant velocity.
2. Column A should be "Position," and Column B should be "Time.": This would imply time changes with position, less intuitive for this data.
3. Column A should be "Velocity," and Column B should be "Speed.": This doesn't align well with the linear relationship and absolute values given.
4. Column A should be "Speed," and Column B should be "Velocity": Similar to above, this doesn't match the provided data context.
Thus, the best fit titles are:
Column A should be "Time," and Column B should be "Position."
This matches well with the slope and intercept obtained, indicating a typical position change over time relationship.
