The windshield wiper of a bus consists of an arm $\overline{AB}$ and a blade $\overline{BC}$. The arm is $6$ feet long and pivots back and forth around point $A$, sweeping out an arc of $120^\circ$ that is symmetric about the vertical line through $A$. The blade is attached to the arm at its end $B$ and always stays vertical, extending $5$ feet above $B$. What is the area of the windshield cleaned by the wiper, in square feet? (Assume that the windshield is a flat vertical surface.)

1. Put $A$ at the origin. The arm makes $60^\circ$ with the vertical at the extreme positions, so the horizontal coordinate of $B$ moves from $-6\sin60^\circ=-3\sqrt3$ to $3\sqrt3$, a width of $6\sqrt3$.
2. The blade is always vertical, so at each moment it covers one vertical segment of length $5$ starting at $B$, and different positions of the blade cover different vertical lines. Hence over each $x$ with $|x|\le3\sqrt3$ the cleaned region contains exactly one vertical segment of length $5$.
3. The area is $5\cdot6\sqrt3=\boxed{30\sqrt3}$ square feet.