To determine which table has a constant of proportionality between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] of 10, we need to check whether the ratio [tex]\(\frac{y}{x}\)[/tex] is equal to 10 for every pair of [tex]\( (x, y) \)[/tex] in that table.
Let's examine each table step by step.
### Table A
[tex]\[
\begin{array}{|cc|}
\hline
x & y \\
\hline
2 & 20 \\
12 & 132 \\
22 & 220 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[
\frac{20}{2} = 10
\][/tex]
[tex]\[
\frac{132}{12} = 11
\][/tex]
[tex]\[
\frac{220}{22} = 10
\][/tex]
Since one of the ratios (132/12) is not equal to 10, Table A does not have a constant of proportionality of 10.
### Table B
[tex]\[
\begin{array}{|cc|}
\hline
x & y \\
\hline
5 & 20 \\
7 & 30 \\
10 & 40 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[
\frac{20}{5} = 4
\][/tex]
[tex]\[
\frac{30}{7} \approx 4.2857
\][/tex]
[tex]\[
\frac{40}{10} = 4
\][/tex]
None of the ratios in Table B is equal to 10, so Table B does not have the required constant of proportionality.
### Table C
[tex]\[
\begin{array}{|cc|}
\hline
x & y \\
\hline
9 & 90 \\
14 & 140 \\
24 & 240 \\
\hline
\end{array}
\][/tex]
Calculate [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[
\frac{90}{9} = 10
\][/tex]
[tex]\[
\frac{140}{14} = 10
\][/tex]
[tex]\[
\frac{240}{24} = 10
\][/tex]
All the ratios in Table C are equal to 10, so Table C has a constant of proportionality of 10.
Therefore, the correct answer is:
(C)
[tex]\[
\begin{array}{|cc|}
\hline
x & y \\
\hline
9 & 90 \\
14 & 140 \\
24 & 240 \\
\hline
\end{array}
\][/tex]
