To find the value of \(F\), we need to consider the horizontal components of the forces, as the resultant is perpendicular to \(F\).

The horizontal component of the \(20 \text{ N}\) force is \(20 \cos 60^\circ\).

The horizontal component of the \(30 \text{ N}\) force is \(30 \cos 60^\circ\).

The horizontal component of the \(F \text{ N}\) force is \(-F\) (since it acts to the left).

Since the resultant is perpendicular to \(F\), the sum of the horizontal components must be zero:

\(20 \cos 60^\circ + 30 \cos 60^\circ - F = 0\)

\(F = 20 \cos 60^\circ + 30 \cos 60^\circ\)

\(F = 20 \times \frac{1}{2} + 30 \times \frac{1}{2}\)

\(F = 10 + 15\)

\(F = 25\)