The square of [tex]\( p \)[/tex] varies inversely as [tex]\( q \)[/tex], which can be expressed mathematically as:
[tex]\[ p^2 \cdot q = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
First, let's determine the constant [tex]\( k \)[/tex] using the given values [tex]\( q = 6 \)[/tex] and [tex]\( p = 4 \)[/tex]:
[tex]\[ (4)^2 \cdot 6 = k \][/tex]
[tex]\[ 16 \cdot 6 = k \][/tex]
[tex]\[ k = 96 \][/tex]
Now that we know [tex]\( k = 96 \)[/tex], we can find the value of [tex]\( q \)[/tex] when [tex]\( p = 12 \)[/tex]. Using the relationship [tex]\( p^2 \cdot q = k \)[/tex]:
[tex]\[ (12)^2 \cdot q = 96 \][/tex]
[tex]\[ 144 \cdot q = 96 \][/tex]
To find [tex]\( q \)[/tex], we solve for [tex]\( q \)[/tex]:
[tex]\[ q = \frac{96}{144} \][/tex]
[tex]\[ q = \frac{2}{3} \][/tex]
Therefore, the value of [tex]\( q \)[/tex] when [tex]\( p = 12 \)[/tex] is:
[tex]\[ \boxed{\frac{2}{3}} \][/tex]
So, the correct answer is [tex]\( \boxed{\frac{2}{3}} \)[/tex].
