To find the constant of proportionality when the variable \( z \) varies directly with \( \frac{x}{y^2} \), we can express this relationship mathematically as:
\[ z = k \cdot \frac{x}{y^2} \]
where \( k \) is the constant of proportionality.
We are given that \( z = 8 \), \( x = 5 \), and \( y = 2.5 \). First, we need to find \( y^2 \):
\[ y^2 = (2.5)^2 = 6.25 \]
Now, we can substitute the known values into the equation:
\[ 8 = k \cdot \frac{5}{6.25} \]
Next, we calculate \( \frac{5}{6.25} \):
\[ \frac{5}{6.25} = \frac{5}{\frac{25}{4}} = 5 \cdot \frac{4}{25} = \frac{20}{25} = \frac{4}{5} \]
Now our equation becomes:
\[ 8 = k \cdot \frac{4}{5} \]
To find \( k \), we isolate it by multiplying both sides by the reciprocal of \( \frac{4}{5} \):
\[ k = 8 \cdot \frac{5}{4} = 10 \]
So, the constant of proportionality \( k \) is:
\[ \boxed{10} \]