To solve for the constant of variation in cases where one quantity varies directly with one variable and inversely with another, we can use the given information to set up an equation and solve for the constant.
Given that [tex]\( h \)[/tex] varies directly with [tex]\( w \)[/tex] and inversely with [tex]\( p \)[/tex]:
[tex]\[ h = k \frac{w}{p} \][/tex]
Here, [tex]\(k\)[/tex] is the constant of variation we need to find. We have specific values provided:
- [tex]\( w = 4 \)[/tex]
- [tex]\( p = 6 \)[/tex]
- [tex]\( h = 2 \)[/tex]
Substitute these values into the equation:
[tex]\[ 2 = k \frac{4}{6} \][/tex]
Now, simplify the fraction on the right side:
[tex]\[ 2 = k \frac{2}{3} \][/tex]
To isolate [tex]\( k \)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{2}{3}\)[/tex], which is [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ k = 2 \times \frac{3}{2} \][/tex]
Calculate the right side:
[tex]\[ k = 3 \][/tex]
Therefore, the constant of variation is:
[tex]\[
\boxed{3}
\][/tex]
