To find the constant of variation [tex]\( k \)[/tex], we start with the relationship given in the problem. Since [tex]\( r \)[/tex] varies directly with [tex]\( p \)[/tex] and inversely with the product of [tex]\( s \)[/tex] and [tex]\( t \)[/tex], we can write this relationship as:
[tex]\[ r = k \cdot \frac{p}{s \cdot t} \][/tex]
Given the values:
- [tex]\( p = 12 \)[/tex]
- [tex]\( t = 2 \)[/tex]
- [tex]\( s = \frac{1}{6} \)[/tex]
- [tex]\( r = 18 \)[/tex]
Our goal is to find [tex]\( k \)[/tex].
First, compute the product of [tex]\( s \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[ s \cdot t = \frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3} \][/tex]
Now, substitute the known values into the equation:
[tex]\[ 18 = k \cdot \frac{12}{\frac{1}{3}} \][/tex]
To simplify [tex]\( \frac{12}{\frac{1}{3}} \)[/tex]:
[tex]\[ \frac{12}{\frac{1}{3}} = 12 \times 3 = 36 \][/tex]
Now, the equation is:
[tex]\[ 18 = k \cdot 36 \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{18}{36} = \frac{1}{2} \][/tex]
Therefore, the constant of variation [tex]\( k \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
So, the correct answer is [tex]\( \frac{1}{2} \)[/tex].
