AMC 10 · 2012 · #2
Grade 7 geometry-2d
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Inscribed means the circle touches the top and bottom edges, so the width equals the diameter: 2 × 5 = 10.
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?
The two walls keep a constant gap everywhere, so that gap is one fixed length.
▸ Why?
Every point of the circle sits one radius from the centre, so its widest span is two radii.
Use the ratio to get the length
A 2:1 ratio means the length is double the width, so the length is 2 × 10 = 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
Area is length times width: 20 × 10 = 200, which is 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