Answer :
To determine the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we need to analyze how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] changes.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 3 & 4 & 6 & 8 & 12 \\ \hline y & 16 & 12 & 8 & 6 & 4 \\ \hline \end{array} \][/tex]
One way to understand this is by finding the product of each pair [tex]\( (x, y) \)[/tex]. If [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], then the product [tex]\( x \cdot y \)[/tex] should be constant.
Let's calculate the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for each pair:
1. For [tex]\( x = 3 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[ 3 \cdot 16 = 48 \][/tex]
2. For [tex]\( x = 4 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[ 4 \cdot 12 = 48 \][/tex]
3. For [tex]\( x = 6 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[ 6 \cdot 8 = 48 \][/tex]
4. For [tex]\( x = 8 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[ 8 \cdot 6 = 48 \][/tex]
5. For [tex]\( x = 12 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 12 \cdot 4 = 48 \][/tex]
Since the product [tex]\( x \cdot y \)[/tex] is constant (48) for all pairs, we can conclude that [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex].
Therefore, the correct statement of variation is:
C. [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 3 & 4 & 6 & 8 & 12 \\ \hline y & 16 & 12 & 8 & 6 & 4 \\ \hline \end{array} \][/tex]
One way to understand this is by finding the product of each pair [tex]\( (x, y) \)[/tex]. If [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], then the product [tex]\( x \cdot y \)[/tex] should be constant.
Let's calculate the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for each pair:
1. For [tex]\( x = 3 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[ 3 \cdot 16 = 48 \][/tex]
2. For [tex]\( x = 4 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[ 4 \cdot 12 = 48 \][/tex]
3. For [tex]\( x = 6 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[ 6 \cdot 8 = 48 \][/tex]
4. For [tex]\( x = 8 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[ 8 \cdot 6 = 48 \][/tex]
5. For [tex]\( x = 12 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 12 \cdot 4 = 48 \][/tex]
Since the product [tex]\( x \cdot y \)[/tex] is constant (48) for all pairs, we can conclude that [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex].
Therefore, the correct statement of variation is:
C. [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex]