AMC 10 · 2012 · #2
Grade 7 geometry-2d
Pick an answer.
The picture is the key. Tool #1 (Draw a Diagram) turns the word "inscribed" into a measurable fact: the circle is tangent to the top and bottom edges, so the vertical gap it spans — its diameter — is exactly the width of the rectangle. Once the width is pinned down, Tool #7 (Identify Subproblems) finishes in two small steps: get the length from the 2:1 ratio, then multiply length by width for the area.
Read the width off the circle
The short side is the diameter, ten.
A circle squeezed between two parallel walls fills exactly one diameter of space, so that diameter is the gap between the walls.
A circle squeezed between two parallel walls fills exactly one diameter of space.
▸ Why?
Every point of the circle sits one radius from its centre, so the widest span is two radii.
▸ Why?
Parallel walls keep a constant gap everywhere, so that gap is what the circle has to match.
Use the ratio to get the length
The ratio makes the long side 20.
A 2:1 ratio just says the first amount is double the second, so scale the known side up by 2.
6.RP.A.1Identify SubproblemsMultiply length by width
Multiplying gives 200, choice (E).
Area of a rectangle counts the unit squares inside, and that is always length times width.
4.MD.A.3Identify SubproblemsAn inscribed circle stretches exactly one diameter across, so its diameter is the box's short side — find that first, then the rest is just ratio and area.
- Read the width off the circle
- Use the ratio to get the length
- Multiply length by width