IDNLearn.com is your go-to resource for finding expert answers and community support. Get the information you need quickly and accurately with our reliable and thorough Q&A platform.
Sagot :
To determine which table has a constant of proportionality between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] of [tex]\(\frac{3}{4}\)[/tex], we need to check if the ratio [tex]\(\frac{y}{x}\)[/tex] is consistently [tex]\(\frac{3}{4}\)[/tex] for all pairs [tex]\((x, y)\)[/tex] in the table. Let’s check each table step-by-step:
Table A:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 8 & 6 \\ 9 & \frac{27}{4} \\ 10 & \frac{15}{2} \\ \hline \end{array} \][/tex]
1. For [tex]\( (8, 6) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{6}{8} = \frac{3}{4} \][/tex]
2. For [tex]\( (9, \frac{27}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{27}{4}}{9} = \frac{27}{4} \times \frac{1}{9} = \frac{27}{36} = \frac{3}{4} \][/tex]
3. For [tex]\( (10, \frac{15}{2}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{15}{2}}{10} = \frac{15}{2} \times \frac{1}{10} = \frac{15}{20} = \frac{3}{4} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table A is [tex]\(\frac{3}{4}\)[/tex], Table A has a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Table B:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 3 & \frac{3}{4} \\ 4 & 1 \\ 5 & \frac{5}{4} \\ \hline \end{array} \][/tex]
1. For [tex]\( (3, \frac{3}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{3}{4}}{3} = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \][/tex]
2. For [tex]\( (4, 1) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1}{4} = \frac{1}{4} \][/tex]
3. For [tex]\( (5, \frac{5}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{5}{4}}{5} = \frac{5}{4} \times \frac{1}{5} = \frac{5}{20} = \frac{1}{4} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table B is [tex]\(\frac{1}{4}\)[/tex], Table B does not have a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Table C:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 10 & \frac{19}{2} \\ 11 & \frac{41}{4} \\ 12 & 11 \\ \hline \end{array} \][/tex]
1. For [tex]\( (10, \frac{19}{2}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{19}{2}}{10} = \frac{19}{2} \times \frac{1}{10} = \frac{19}{20} \][/tex]
2. For [tex]\( (11, \frac{41}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{41}{4}}{11} = \frac{41}{4} \times \frac{1}{11} = \frac{41}{44} \][/tex]
3. For [tex]\( (12, 11) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{11}{12} = \frac{11}{12} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table C is different and not equal to [tex]\(\frac{3}{4}\)[/tex], Table C does not have a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Given this analysis, the table that has a constant of proportionality between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] of [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ \boxed{A} \][/tex]
Table A:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 8 & 6 \\ 9 & \frac{27}{4} \\ 10 & \frac{15}{2} \\ \hline \end{array} \][/tex]
1. For [tex]\( (8, 6) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{6}{8} = \frac{3}{4} \][/tex]
2. For [tex]\( (9, \frac{27}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{27}{4}}{9} = \frac{27}{4} \times \frac{1}{9} = \frac{27}{36} = \frac{3}{4} \][/tex]
3. For [tex]\( (10, \frac{15}{2}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{15}{2}}{10} = \frac{15}{2} \times \frac{1}{10} = \frac{15}{20} = \frac{3}{4} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table A is [tex]\(\frac{3}{4}\)[/tex], Table A has a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Table B:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 3 & \frac{3}{4} \\ 4 & 1 \\ 5 & \frac{5}{4} \\ \hline \end{array} \][/tex]
1. For [tex]\( (3, \frac{3}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{3}{4}}{3} = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \][/tex]
2. For [tex]\( (4, 1) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1}{4} = \frac{1}{4} \][/tex]
3. For [tex]\( (5, \frac{5}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{5}{4}}{5} = \frac{5}{4} \times \frac{1}{5} = \frac{5}{20} = \frac{1}{4} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table B is [tex]\(\frac{1}{4}\)[/tex], Table B does not have a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Table C:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 10 & \frac{19}{2} \\ 11 & \frac{41}{4} \\ 12 & 11 \\ \hline \end{array} \][/tex]
1. For [tex]\( (10, \frac{19}{2}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{19}{2}}{10} = \frac{19}{2} \times \frac{1}{10} = \frac{19}{20} \][/tex]
2. For [tex]\( (11, \frac{41}{4}) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{\frac{41}{4}}{11} = \frac{41}{4} \times \frac{1}{11} = \frac{41}{44} \][/tex]
3. For [tex]\( (12, 11) \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{11}{12} = \frac{11}{12} \][/tex]
Since each [tex]\(\frac{y}{x}\)[/tex] value in Table C is different and not equal to [tex]\(\frac{3}{4}\)[/tex], Table C does not have a constant proportionality of [tex]\(\frac{3}{4}\)[/tex].
Given this analysis, the table that has a constant of proportionality between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] of [tex]\(\frac{3}{4}\)[/tex] is:
[tex]\[ \boxed{A} \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.