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2010 AMC 8

Problem 18

Problem 18 · 2010 AMC 8 Medium
Geometry & Measurement area-ratio

A decorative window is made up of a rectangle with semicircles on either end. The ratio of AD to AB is 3 : 2, and AB is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

Figure for AMC 8 2010 Problem 18
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Answer: C — 6 : π.
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Hint 1 of 2
The two semicircles share the same diameter (the short side, 30). Slide them together and they make exactly one full circle — so stop treating them as two shapes.
Still stuck? Show hint 2 →
Hint 2 of 2
Two equal semicircles always glue into one whole circle. Replacing ‘two halves’ with ‘one circle’ is the move that makes the area easy.
Show solution
Approach: glue the two semicircles into one circle
  1. Both semicircles sit on a side of length AB = 30, so glued together they form one circle of diameter 30 — radius 15, area π · 152 = 225π.
  2. The rectangle is AD × AB = (3/2 · 30) × 30 = 45 × 30 = 1350.
  3. Ratio rectangle : circle = 1350 : 225π. Divide both by 225: 6 : π.
  4. Why this transfers: matched semicircles (or quarter-circles at four corners) recombine into whole circles — spotting that turns a scary ‘sum of curved pieces’ into a single πr2. The 30:π trap comes from forgetting to square the radius.
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