AMC 10 · 2004 · #19
Grade 7 geometry-2dA white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
Pick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A cylindrical silo is $30$ feet across and $80$ feet tall. A red stripe with a horizontal width of $3$ feet spirals up the outside, wrapping all the way around the silo two full times as it climbs from bottom to top. Find the area of the stripe.
Givens: The silo is a cylinder with diameter $30$ feet and height $80$ feet.; The red stripe has a horizontal width of $3$ feet.; The stripe makes two complete revolutions around the silo as it goes from the bottom to the top.; Answer choices (in square feet): (A) $120$, (B) $180$, (C) $240$, (D) $360$, (E) $480$.
Unknowns: The area of the red stripe, in square feet.
Understand
Restated: A cylindrical silo is $30$ feet across and $80$ feet tall. A red stripe with a horizontal width of $3$ feet spirals up the outside, wrapping all the way around the silo two full times as it climbs from bottom to top. Find the area of the stripe.
Givens: The silo is a cylinder with diameter $30$ feet and height $80$ feet.; The red stripe has a horizontal width of $3$ feet.; The stripe makes two complete revolutions around the silo as it goes from the bottom to the top.; Answer choices (in square feet): (A) $120$, (B) $180$, (C) $240$, (D) $360$, (E) $480$.
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #9 Solve an Easier Related Problem, #1 Draw a Diagram
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.
Execute — Answer: C
7.G.B.6 Step 1 Slit the silo and roll it flat
- The stripe lies on the curved side of the silo, which is hard to measure as-is.
- Imagine cutting the silo straight down along one vertical line and unrolling its side onto a table, like peeling the label off a can and laying it flat.
- Flattening a surface this way does not stretch or squash it, so every area stays exactly the same.
- What you get is a flat rectangle that is $80$ feet tall — the height of the silo.
💡 Rolling a curved surface out flat without stretching keeps every area unchanged, turning a hard curved shape into an easy flat one.
6.G.A.1 Step 2 The spiral becomes a slanted band
- As the stripe climbs, it also slides sideways around the silo, so on the flat sheet it turns into a band that leans over — a parallelogram.
- But its width never changes.
- "Horizontal width $3$ feet" means that if you slice across the stripe level, at any height, the slice is always $3$ feet long.
- So the leaning band is $3$ feet wide the whole way up the $80$-foot sheet.
💡 The stripe is a parallelogram: at every height it is the same $3$ feet wide, just nudged sideways as it rises.
6.G.A.1 Step 3 Straighten it into a 3-by-80 rectangle
- A parallelogram covers the same area as the rectangle with the same base and the same height — sliding its level slices sideways to line them up never adds or removes any area.
- So the slanted stripe covers exactly as much as a stripe that runs straight up with no spiralling: a rectangle $3$ feet wide and $80$ feet tall.
- This is the reason the two revolutions do not matter at all.
💡 Sliding a stack of equal-width level slices sideways packs them into a plain rectangle without changing the total area.
4.NBT.B.5 Step 4 Multiply to get the area
- The area is the width times the height: $3 \times 80 = 240$ square feet.
- Notice the diameter of $30$ feet and the two revolutions never entered the arithmetic — they were placed there to distract.
- The area of the stripe is $240$ square feet, which is choice (C).
💡 Once the stripe is a plain rectangle, its area is simply width times height.
7.G.B.6 The stripe lies on the curved side of the silo, which is hard to measure as-is. 6.G.A.1 As the stripe climbs, it also slides sideways around the silo, so on the flat sh 6.G.A.1 A parallelogram covers the same area as the rectangle with the same base and the 4.NBT.B.5 The area is the width times the height: $3 \times 80 = 240$ square feet. Notice Review
Reasonableness: The stripe is $3$ feet wide and runs the full $80$ feet of height, so $240$ square feet — just $3 \times 80$ — is a sensible size, a thin band using a small part of the silo's side. The tempting wrong answers all come from misusing the distractor numbers: (E) $480$ is $240$ doubled, as if the two revolutions doubled the area; (A) $120$ is $240$ halved; and (D) $360$ comes from multiplying $3$ by a slant length instead of the true vertical height $80$. The width-times-height reasoning ignores the slant and the diameter and lands cleanly on (C) $240$.
Alternative: Think of the stripe as a tall stack of very thin horizontal strips. At every height the strip is $3$ feet wide, and the strips stack through all $80$ feet from bottom to top. Whether each thin strip sits straight above the one below (a straight stripe) or is nudged sideways (the spiral stripe), the strips have the very same total area, because nudging a strip sideways does not change its size. Adding them up gives $3 \times 80 = 240$ square feet again.
CCSS standards used (min grade 7)
7.G.B.6Solve real-world problems involving area, surface area, and volume (Unrolling the silo's curved side into a flat rectangle without changing any area, so the stripe can be measured on a flat surface.)6.G.A.1Find area of triangles, special quadrilaterals, and polygons by composing and decomposing (Seeing the flattened stripe as a parallelogram and using that it has the same area as a rectangle with the same base ($3$) and height ($80$).)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Computing the rectangle's area, $3 \times 80 = 240$ square feet.)
⭐ Unroll 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.
⭐ Unroll 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.
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