To determine which equation can be used to find any corresponding length [tex]\(l\)[/tex] and width [tex]\(w\)[/tex] that fit the pattern in the given table, we need to analyze the relationship between [tex]\(l\)[/tex] and [tex]\(w\)[/tex].
The table provides the following data pairs for width ([tex]\(w\)[/tex]) and length ([tex]\(l\)[/tex]):
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Width} \, (w) & \text{Length} \, (l) \\
\hline
2 & 37.5 \\
\hline
4 & 18.75 \\
\hline
6 & 12.5 \\
\hline
8 & 9.375 \\
\hline
\end{array}
\][/tex]
We hypothesize that the relationship between [tex]\(w\)[/tex] and [tex]\(l\)[/tex] follows the equation [tex]\(l = \frac{k}{w}\)[/tex], where [tex]\(k\)[/tex] is a constant.
To verify this, we need to calculate the product [tex]\(l \cdot w\)[/tex] for each pair of values provided in the table and check if the product remains constant.
[tex]\[
\begin{align*}
(2, 37.5) &: 2 \cdot 37.5 = 75.0 \\
(4, 18.75) &: 4 \cdot 18.75 = 75.0 \\
(6, 12.5) &: 6 \cdot 12.5 = 75.0 \\
(8, 9.375) &: 8 \cdot 9.375 = 75.0 \\
\end{align*}
\][/tex]
Since the product [tex]\(l \cdot w\)[/tex] is 75.0 for each pair, it confirms that [tex]\(k = 75.0\)[/tex]. This means the relationship between the width [tex]\(w\)[/tex] and the length [tex]\(l\)[/tex] is indeed described by the equation [tex]\(l = \frac{k}{w}\)[/tex], with [tex]\(k = 75.0\)[/tex].
Thus, the correct equation to represent the relationship between the width and the length of the rectangle, given the data in the table, is:
[tex]\[ l = \frac{75.0}{w} \][/tex]
Therefore, the equation [tex]\(l = \frac{k}{w}\)[/tex] fits the pattern in the table, where [tex]\(k = 75.0\)[/tex].
