The inverse variation relationship between \(P\) and \(\sqrt{q}\) can be expressed as \(P \propto \frac{1}{\sqrt{q}}\).

To find the constant of variation, we can set up the equation:

\(P_1 \cdot \sqrt{q_1} = P_2 \cdot \sqrt{q_2}\), where \(P_1\), \(q_1\) are the initial values and \(P_2\), \(q_2\) are the final values.

Given that P = 3 when q = 16, and we need to find q when P = 4. Substituting the values into the equation:

\(3 \cdot \sqrt{16} = 4 \cdot \sqrt{q}\)

\(3 \cdot 4 = 4 \cdot \sqrt{q}\)

\(12 = 4 \cdot \sqrt{q}\)

Divide both sides by 4:

\(\sqrt{q} = 3\)

Squaring both sides:

\(q = 3^2\)

q = 9