To determine which column of distance data represents a quadratic relationship, let's analyze the given data:
1. Understanding a Quadratic Relationship:
- A quadratic relationship means that the distance [tex]\( d \)[/tex] is proportional to the square of time [tex]\( t \)[/tex].
- Mathematically, this can be expressed as [tex]\( d = k \cdot t^2 \)[/tex], where [tex]\( k \)[/tex] is a constant.
2. Analyzing Column A:
- Given data for time [tex]\( t \)[/tex] and distance [tex]\( d \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Time (s)} & \text{A. Distance (m)} \\
\hline
0 & 0 \\
\hline
1 & 1 \\
\hline
2 & 4 \\
\hline
3 & 9 \\
\hline
4 & 16 \\
\hline
5 & 25 \\
\hline
6 & 36 \\
\hline
\end{tabular}
\][/tex]
- Let's check if this follows [tex]\( d = k \cdot t^2 \)[/tex]:
[tex]\[
\begin{align*}
t = 0 & , \quad d = 0 = 0^2 \\
t = 1 & , \quad d = 1 = 1^2 \\
t = 2 & , \quad d = 4 = 2^2 \\
t = 3 & , \quad d = 9 = 3^2 \\
t = 4 & , \quad d = 16 = 4^2 \\
t = 5 & , \quad d = 25 = 5^2 \\
t = 6 & , \quad d = 36 = 6^2 \\
\end{align*}
\][/tex]
- Clearly, distance [tex]\( d \)[/tex] is following a quadratic relationship [tex]\( d = t^2 \)[/tex].
3. Analyzing Column B:
- Given data for time [tex]\( t \)[/tex] and distance [tex]\( d \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Time (s)} & \text{B. Distance (m)} \\
\hline
0 & 2 \\
\hline
1 & 4 \\
\hline
2 & 6 \\
\hline
3 & 8 \\
\hline
4 & 10 \\
\hline
5 & 12 \\
\hline
6 & 14 \\
\hline
\end{tabular}
\][/tex]
- This data does not follow the quadratic form [tex]\( d = k \cdot t^2 \)[/tex]. It appears to follow a linear relationship instead.
4. Analyzing Column C:
- Given data for time [tex]\( t \)[/tex] and distance [tex]\( d \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Time (s)} & \text{C. Distance (m)} \\
\hline
0 & 9 \\
\hline
1 & 18 \\
\hline
2 & 27 \\
\hline
3 & 36 \\
\hline
4 & 45 \\
\hline
5 & 54 \\
\hline
6 & 63 \\
\hline
\end{tabular}
\][/tex]
- Checking for quadratic relationship:
[tex]\[
\begin{align*}
t = 0 & , \quad d = 9 \neq 0^2 \\
t = 1 & , \quad d = 18 \neq 1^2 \\
t = 2 & , \quad d = 27 \neq 2^2 \\
t = 3 & , \quad d = 36 \neq 3^2 \\
t = 4 & , \quad d = 45 \neq 4^2 \\
t = 5 & , \quad d = 54 \neq 5^2 \\
t = 6 & , \quad d = 63 \neq 6^2 \\
\end{align*}
\][/tex]
- Clearly, this distance data does not follow a quadratic relationship.
Conclusion:
- Column A represents a quadratic relationship as distance [tex]\( d \)[/tex] is given by [tex]\( d = t^2 \)[/tex].
Thus, the correct answer is:
A. Column A
