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The distance versus time plot for a particular object shows a quadratic relationship. Which column of distance data is possible for this situation?

\begin{tabular}{|l|l|l|l|l|l|}
\hline Time [tex]$( s )$[/tex] & A. Distance [tex]$( m )$[/tex] & B. Distance [tex]$( m )$[/tex] & C. Distance [tex]$( m )$[/tex] & D. Distance [tex]$( m )$[/tex] & E. Distance [tex]$( m )$[/tex] \\
\hline 0 & 0 & 2.00 & 9.00 & [tex]$\ddagger$[/tex] & [tex]$\ddagger$[/tex] \\
\hline 1 & 1.00 & 4.00 & 18.00 & 1.00 & 1.00 \\
\hline 2 & 4.00 & 6.00 & 27.00 & 0.50 & 0.25 \\
\hline 3 & 9.00 & 8.00 & 36.00 & 0.33 & 0.11 \\
\hline 4 & 16.00 & 10.00 & 45.00 & 0.25 & 0.06 \\
\hline 5 & 25.00 & 14.00 & 54.00 & 0.20 & 0.04 \\
\hline 6 & 36.00 & 63.00 & 0.16 & 0.02 \\
\hline
\end{tabular}

A. column [tex]$A$[/tex]
B. column [tex]$B$[/tex]
C. column [tex]$C$[/tex]
D. column [tex]$D$[/tex]
E. column [tex]$E$[/tex]



Answer :

GinnyAnswer
To identify which column of distance data shows a quadratic relationship with time, we need to carefully examine the given distance values and their progression over time. A quadratic relationship suggests that the distance increases in a manner proportional to the square of the time, i.e., [tex]\( d(t) = at^2 + bt + c \)[/tex].

Let's analyze each column:

1. Column A:
- At [tex]\( t = 0 \)[/tex], [tex]\( d = 0 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( d = 1 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( d = 4 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( d = 9 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( d = 16 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( d = 25 \)[/tex]

The pattern here is [tex]\( d(t) = t^2 \)[/tex], which matches the quadratic form perfectly.

2. Column B:
- At [tex]\( t = 0 \)[/tex], [tex]\( d = 2 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( d = 4 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( d = 6 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( d = 8 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( d = 10 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( d = 14 \)[/tex]

This column shows a linear increment and does not match a quadratic pattern.

3. Column C:
- At [tex]\( t = 0 \)[/tex], [tex]\( d = 9 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( d = 18 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( d = 27 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( d = 36 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( d = 45 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( d = 54 \)[/tex]

The values here are consistent with [tex]\( d(t) = 9t \)[/tex], indicating a linear relationship, not a quadratic one.

4. Column D:
- At [tex]\( t = 0 \)[/tex], [tex]\( d = \text{undefined (nan)} \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( d = 1 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( d = 0.5 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( d = 0.33 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( d = 0.25 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( d = 0.20 \)[/tex]

This column sees diminishing returns and does not resemble a quadratic relationship.

5. Column E:
- At [tex]\( t = 0 \)[/tex], [tex]\( d = \text{undefined (nan)} \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( d = 1 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( d = 0.25 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( d = 0.11 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( d = 0.06 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( d = 0.04 \)[/tex]

The pattern of values here indicates a rapid decrease and does not match a quadratic progression.

Only Column A fits the described quadratic relationship [tex]\( d(t) = t^2 \)[/tex].

Therefore, the correct answer is:
- A. column A

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