AMC 12 2025b Problem 12

The windshield wiper on the driver's side of a large bus is depicted below.

/* Made by MRENTHUSIASM */
size(200);
pair A, B, C, D;
A = origin;
B = (3/2,3*sqrt(3)/2);
C = B+(0,1.75);
D = B-(0,1.75);
draw(A--B,linewidth(1.1));
draw(C--D,red+linewidth(2.1));
dot(A^^B,linewidth(5));
label("$A$",A,1.5*W);
label("$B$",B,1.5*E);
label("$C$",C,1.5*E);
label("$D$",D,1.5*E);

Arm $\overline{AB}$ pivots back and forth around point $A$, sweeping out an arc of $60^{\circ}$, symmetric about the vertical line through $A$. The wiper blade $\overline{CD}$ is attached to $B$ at its midpoint and stays vertical as the arm moves. The arm is $3$ feet long, and the wiper blade is $3.5$ feet tall. What is the area of the windshield cleaned by the wiper, in square feet, to the nearest hundredth? (Assume that the windshield is a flat vertical surface.)

$\textbf{(A) } 9.68 \textbf{(B) } 10.14 \textbf{(C) } 10.50 \textbf{(D) } 11.32 \textbf{(E) } 12.00$