To solve the problem of finding the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 8, given that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is 6 when [tex]\( x \)[/tex] is 72, we can follow these steps:
1. Understand the relationship: Since [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], we can express this relationship as [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
2. Determine the constant of proportionality [tex]\( k \)[/tex]:
We know that [tex]\( y = 6 \)[/tex] when [tex]\( x = 72 \)[/tex]. Using this information, we can solve for [tex]\( k \)[/tex].
[tex]\[
6 = k \cdot 72
\][/tex]
[tex]\[
k = \frac{6}{72} = \frac{1}{12}
\][/tex]
3. Use the constant of proportionality [tex]\( k \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 8 \)[/tex]:
Now that we know [tex]\( k = \frac{1}{12} \)[/tex], we can use this constant to find [tex]\( y \)[/tex] when [tex]\( x = 8 \)[/tex].
[tex]\[
y = k \cdot x = \frac{1}{12} \cdot 8 = \frac{8}{12} = \frac{2}{3}
\][/tex]
Therefore, the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 8 is [tex]\( \frac{2}{3} \)[/tex].
So the correct answer is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
