Solution:

The answer is (B). Let $f$ be the road on which Eva is located and $s$ the one on which the lampposts are located. Consider the circle passing through $P, Q, E$, which exists since $P, Q$ belong to $s$ while $E$ does not belong to it, otherwise the angle $P \hat{E} Q$ would be 0, and suppose by contradiction that it is not tangent to $f$. Then there exists the further intersection point $E'$ of this circle with $f$, and let $E''$ be a point on the arc $E E'$ not containing $P$ and $Q$; we have $P \hat{E} Q = P \hat{E}'' Q$ since they subtend the same arc $P Q$; let $D$ be the intersection of $f$ and $P E''$, then $P \hat{D} Q > P \hat{E}'' Q = P \hat{E} Q$ since it is an exterior angle of the triangle $P D E''$ not adjacent to the angle at $E''$. Then the point $E$ could not give the maximum angle, and therefore the circle through $P, Q, E$ must necessarily be tangent to $f$ at $E$. But then, by the tangent-secant theorem, $L E^2 = L P \cdot L Q$, hence $L E = \sqrt{40 \cdot 90} = 60$.

(Kuzmin)