2025 AMC 12B Problem 12

Question

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

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.)

9.68

10.14

10.50

11.32

12.00

Answer: C

Concepts:area arc

Difficulty rating: 1690

Small hint:

As the arm turns,

B

traces an arc of radius

3;

the blade is always a vertical segment of height

3.5

centered at

B

Big hint:

Each vertical line meets the cleaned region in a segment of constant height

3.5,

so the area is that height times the horizontal width swept

Solution:

Put

A

at the origin. Then

B = (3\sin\theta, 3\cos\theta)

for

\theta \in [-30^\circ, 30^\circ],

so the horizontal coordinate of

B

ranges over

[-1.5, 1.5],

a width of

3.

At each horizontal position exactly one vertical blade of height

3.5

passes through, so by Cavalieri’s principle the cleaned area is

3.5 \times 3 = 10.5

square feet.

Thus, the correct answer is C.