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A rectangle with constant area has possible lengths and widths as shown in the table below.

Width vs. Length of a Rectangle
\begin{tabular}{|c|c|}
\hline Width [tex]$(w)$[/tex] & Length [tex]$(l)$[/tex] \\
\hline 2 & 37.5 \\
\hline 4 & 18.75 \\
\hline 6 & 12.5 \\
\hline 8 & 9.375 \\
\hline
\end{tabular}

Which equation can be used to find any corresponding length and width that fit the pattern in this table?

A. [tex]$I=\frac{k}{w}$[/tex], where [tex]$l$[/tex] is the length, [tex]$w$[/tex] is the width, and [tex]$k$[/tex] is a constant
B. [tex]$I=m w+b$[/tex], where [tex]$l$[/tex] is the length, [tex]$w$[/tex] is the width, and [tex]$m$[/tex] and [tex]$b$[/tex] are constants
C. [tex]$I=k w^{\frac{1}{2}}$[/tex], where [tex]$l$[/tex] is the length, [tex]$w$[/tex] is the width, and [tex]$k$[/tex] is a constant
D. [tex]$I=a w^2$[/tex], where [tex]$l$[/tex] is the length, [tex]$w$[/tex] is the width, and [tex]$a$[/tex] is a constant



Answer :

GinnyAnswer
Let's analyze the given options to determine the equation that fits the pattern provided in the table.

The table given is:
[tex]\[ \begin{tabular}{|c|c|} \hline Width (w) & Length (I) \\ \hline 2 & 37.5 \\ \hline 4 & 18.75 \\ \hline 6 & 12.5 \\ \hline 8 & 9.375 \\ \hline \end{tabular} \][/tex]

To find the relationship between the width [tex]\( w \)[/tex] and the length [tex]\( l \)[/tex], consider the provided options and test them against the data:

1. Equation: [tex]\( l = \frac{k}{w} \)[/tex]

This implies that the length is inversely proportional to the width. We will test this by checking if the product [tex]\( w \times l \)[/tex] is a constant [tex]\( k \)[/tex]:

[tex]\[ \begin{align*} 2 \times 37.5 &= 75 \\ 4 \times 18.75 &= 75 \\ 6 \times 12.5 &= 75 \\ 8 \times 9.375 &= 75 \\ \end{align*} \][/tex]

Since the product of width and length is consistently 75, the equation [tex]\( l = \frac{k}{w} \)[/tex] fits perfectly with [tex]\( k = 75 \)[/tex].

2. Equation: [tex]\( l = mw + b \)[/tex]

This implies a linear relationship between the width and length. We can test this by plotting the points and checking if they form a straight line. Notice from the data itself, as the width increases, the length decreases. Such a linear equation cannot model the data correctly because it suggests length increases or decreases linearly, which is not represented by the given values.

3. Equation: [tex]\( l = kw^{\frac{1}{2}} \)[/tex]

This implies that the length is proportional to the square root of the width. We can test points to see if:

[tex]\[ \begin{align*} 37.5 &= k \cdot 2^{\frac{1}{2}} \\ 18.75 &= k \cdot 4^{\frac{1}{2}} \\ 12.5 &= k \cdot 6^{\frac{1}{2}} \\ 9.375 &= k \cdot 8^{\frac{1}{2}} \\ \end{align*} \][/tex]

But the values computed do not satisfy the constant [tex]\( k \)[/tex] in each due to the non-linear nature expected by this equation.

4. Equation: [tex]\( l = aw^2 \)[/tex]

This implies that the length is proportional to the square of the width. Testing points:

[tex]\[ \begin{align*} 37.5 &= a \cdot 2^2 \\ 18.75 &= a \cdot 4^2 \\ 12.5 &= a \cdot 6^2 \\ 9.375 &= a \cdot 8^2 \\ \end{align*} \][/tex]

This would also suggest unrealistic values as practical empirical values [tex]\( 37.5, 18.75\)[/tex] scaling quickly beyond to non-proportionate data.

Given the data in the table, the equation that best describes the relationship is:

[tex]\[ l = \frac{k}{w}, \][/tex]

where [tex]\( k \)[/tex] is a constant. From our observations, [tex]\( k = 75 \)[/tex]. Therefore, the correct equation is [tex]\( l = \frac{75}{w} \)[/tex].

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