AMC 10 · 2025 · #12

Grade 10 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
spatial-visualizationarea-rectanglesthirty-sixty-ninety-triangle identify-subproblemsarea-difference ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A wiper arm 3 feet long is hinged at a point A on a flat vertical windshield. The arm swings back and forth through a total angle of 60 degrees, and the swing is symmetric about the vertical line through A, so the arm tilts at most 30 degrees each way. A straight rubber blade 3.5 feet tall is fastened to the far end B of the arm at the blade's midpoint. The blade never turns: it stays vertical the whole time, so it just rides along while B travels on its arc. Find the area of glass the blade touches, in square feet, rounded to the nearest hundredth.

Pick an answer.

(A)
9.68
(B)
10.14
(C)
10.50
(D)
11.32
(E)
12.00
How to solve
Strategy Visualize Spatial Relationships

The whole problem lives or dies on picturing the motion correctly. A wiper you have seen on a car turns its blade with the arm, and that mental image sweeps a curved ring. Here the blade is told to stay vertical, so I should picture a rigid vertical stick being carried sideways along an arc while never tilting. Once I hold that picture, the natural way to measure the swept shape is to cut it with vertical lines rather than to chase its curved boundary, because each vertical line meets the shape in one piece of the blade, and the blade always has the same length. That turns a curved band into a rectangle in disguise, which is an area I can compute in one multiplication.

1STEP 1

Picture the blade riding, not turning

The blade rides without turning.

A=(0,0), AB = 3, 1/2 · 3.5 = 1.75
2STEP 2

Name the tilt angle and locate B

B sweeps horizontally from -1.5 to 1.5.

B(θ) = (3sinθ, 3cosθ), -30^° ≤ θ ≤ 30^°, x_B ∈ [-1.5, 1.5]
3STEP 3

Cut the region with vertical lines

Cut the region with vertical lines.

cleaned region = {(x,y) : -1.5 ≤ x ≤ 1.5, √(9-x²)-1.75 ≤ y ≤ √(9-x²)+1.75}
4STEP 4

Every slice has the same height

Every slice has the same height 3.5.

(√(9-x²)+1.75)-(√(9-x²)-1.75) = 3.5 for every x ∈ [-1.5, 1.5]
5STEP 5

Multiply width by height

Three times 3.5 is 10.50.

Area = 3 × 3.5 = 10.5 = 10.50 square feet
Answer
10.50
The number is believable in size. The blade is 3.5 feet tall and never leaves a strip 3 feet wide, so the cleaned area cannot exceed the area of a box holding that strip, and the box from the lowest sweep point to the highest is 3 feet by about 3.90 feet, or about 11.71 square feet. Our 10.50 sits just under that, as it should for a shape that curves inside the box. The result also explains the trap: if the blade turned with the arm instead of staying vertical, the swept shape would be a 60^° slice of a ring with inner radius 3-1.75=1.25 and outer radius 3+1.75=4.75, giving 1/6π(4.75²-1.25²) = 3.5π ≈ 11.00 square feet, which is not one of the five choices. That mismatch confirms that the phrase 'stays vertical' is doing real work. Finally, the sweep geometry checks out: the two extreme positions of B are 3 feet apart and each is 3 feet from A, so triangle ABB' is equilateral, exactly what a 60^° sweep should produce.
💡Key takeaway

When a shape is swept out by something that slides without turning, slice the shape the same way that thing is shaped, and if every slice is the same length you are really just looking at a rectangle.

  • Picture the blade riding, not turning
  • Name the tilt angle and locate B
  • Cut the region with vertical lines
  • Every slice has the same height
  • Multiply width by height

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