To determine which table represents a proportional relationship with a constant of proportionality equal to 0.8, we need to evaluate each table.
1. Table 1:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 4 & 8 & 10 \\
\hline
y & 0 & 0.5 & 1 & 1.25 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each [tex]\(x \neq 0\)[/tex]:
- For [tex]\(x = 4\)[/tex], [tex]\(\frac{y}{x} = \frac{0.5}{4} = 0.125\)[/tex]
- For [tex]\(x = 8\)[/tex], [tex]\(\frac{y}{x} = \frac{1}{8} = 0.125\)[/tex]
- For [tex]\(x = 10\)[/tex], [tex]\(\frac{y}{x} = \frac{1.25}{10} = 0.125\)[/tex]
The constant of proportionality is 0.125, not 0.8.
2. Table 2:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 5 & 10 & 12.5 \\
\hline
y & 0 & 4 & 8 & 10 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each [tex]\(x \neq 0\)[/tex]:
- For [tex]\(x = 5\)[/tex], [tex]\(\frac{y}{x} = \frac{4}{5} = 0.8\)[/tex]
- For [tex]\(x = 10\)[/tex], [tex]\(\frac{y}{x} = \frac{8}{10} = 0.8\)[/tex]
- For [tex]\(x = 12.5\)[/tex], [tex]\(\frac{y}{x} = \frac{10}{12.5} = 0.8\)[/tex]
The constant of proportionality is indeed 0.8 for Table 2.
3. Table 3:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 4 & 8 & 10 \\
\hline
y & 0.8 & 0.8 & 0.8 & 0.8 \\
\hline
\end{array}
\][/tex]
This table cannot represent a proportional relationship, as when [tex]\(x\)[/tex] changes, [tex]\(y\)[/tex] remains constant.
4. Table 4:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 5 & 10 & 12.5 \\
\hline
y & 0.8 & 10.8 & 20.8 & 25.8 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each [tex]\(x \neq 0\)[/tex]:
- For [tex]\(x = 5\)[/tex], [tex]\(\frac{y}{x} = \frac{10.8}{5} = 2.16\)[/tex]
- For [tex]\(x = 10\)[/tex], [tex]\(\frac{y}{x} = \frac{20.8}{10} = 2.08\)[/tex]
- For [tex]\(x = 12.5\)[/tex], [tex]\(\frac{y}{x} = \frac{25.8}{12.5} = 2.064\)[/tex]
The constant of proportionality is not consistent and is not 0.8.
Therefore, the table that represents a proportional relationship with a constant of proportionality equal to 0.8 is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 5 & 10 & 12.5 \\
\hline
y & 0 & 4 & 8 & 10 \\
\hline
\end{array}
\][/tex]
So, the answer is Table 2.
