AMC 10 · 2004 · #19
Grade 7 geometry-2d
Pick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The stripe sits on the curved side of the silo, which is awkward to measure directly. The key move is to imagine slitting the silo straight down and unrolling its side into a flat sheet 80 feet tall (Tool #17) — rolling a surface out flat does not stretch it, so every area is preserved. On the flat sheet the spiralling stripe becomes a slanted band, a parallelogram, that is still 3 feet wide at every height. Then, instead of wrestling with the slant, compare it to the far easier case of a stripe going straight up with no spiralling — a plain 3-by-80 rectangle (Tool #9). A parallelogram has the same area as the rectangle with the same base and height, so the spiralling, the number of revolutions, and the silo's diameter all turn out to be distractions.
Slit the silo and roll it flat
Slit the silo straight down and unroll its side flat — no stretching, so every area is unchanged. You get a rectangle 80 feet tall.
Rolling a curved surface out flat without stretching keeps every area unchanged, turning a hard curved shape into an easy flat one.
Rolling a curved surface out flat without stretching keeps every area unchanged.
▸ Why?
Unrolling moves the surface without stretching it, so lengths and areas along it survive.
▸ Why?
The rim's length is two pi times the radius, which is exactly the width of the flattened sheet.
The spiral becomes a slanted band
Rising while sliding sideways, the stripe becomes a slanted band — a parallelogram — still 3 feet wide at every height.
The stripe is a parallelogram: at every height it is the same 3 feet wide, just nudged sideways as it rises.
6.G.A.1Visualize Spatial RelationshipsStraighten it into a 3-by-80 rectangle
A parallelogram has the same area as the rectangle with the same base and height — so the stripe matches a plain 3-by-80 rectangle.
Sliding a stack of equal-width level slices sideways packs them into a plain rectangle without changing the total area.
6.G.A.1Solve An Easier Related ProblemMultiply to get the area
Width times height: 3 × 80 = 240 square feet — the diameter 30 and the two revolutions never entered the arithmetic.
Once the stripe is a plain rectangle, its area is simply width times height.
4.NBT.B.5Solve An Easier Related ProblemUnroll a curved surface flat, and a slanted stripe has the same area as a straight one of the same width and height — so just multiply width by height.
- Slit the silo and roll it flat
- The spiral becomes a slanted band
- Straighten it into a 3-by-80 rectangle
- Multiply to get the area
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