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
