AMC 8 · 2024 · #22

Grade 7 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

area-circlesrate dimensional-analysisarea-difference ↑ Prerequisites: area-circlesfraction-decimal-conversion

📏 Medium solution 💡 4 insights 📊 Diagram

Problem

A roll of tape looks like a flat ring (an annulus) when you look at it end-on: the outer disk has diameter 4 in and there's a hole in the middle of diameter 2 in. The tape itself is 0.015 in thick. If you unrolled the whole thing into one long flat strip, about how long would it be (rounded to the nearest 100 inches)?

Pick an answer.

(A)

300

(B)

600

(C)

1200

(D)

1500

(E)

1800

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The key trick is to draw two pictures side by side (Tool #1): the ring-shaped cross-section of the roll, and the same tape unrolled into a long thin rectangle. Imagining the unrolling physically (Tool #10) makes it obvious that the same amount of tape-material is just rearranged — the cross-sectional area doesn't change. Then we break the work into clean subproblems (Tool #7): (a) area of the ring, (b) length from area = length × thickness, (c) round. Finally we use Tool #3 to match the rounded value to a choice. This avoids reaching for algebra (Tool #13); the only formula we truly need is area of a circle.

1STEP 1

Unrolling keeps the tape's cross-section, so the ring's area equals the unrolled strip's area: Area of ring = L × t.

Area_ring = L × t

2STEP 2

Radius is half the diameter, giving R = 2 in, r = 1 in.

R =

\frac{4}{2}

= 2 in, r =

\frac{2}{2}

= 1 in

3STEP 3

The ring's area is the big disk minus the hole: π·2² − π·1² = 3π square inches.

Area_ring = π R² - π r² = π(2²) - π(1²) = 4π - π = 3π

4STEP 4

Since 3π = L × 0.015, divide: L = 3π ÷ 0.015, and with π ≈ 3.14 that's about 628 inches.

L =

\frac{3π}{0.015}

\frac{9.42}{0.015}

= 628 in

5STEP 5

Round 628 to the nearest 100: the tens digit 2 is below 5, so round down to 600 — choice (B).

628 ≈ 600 → (B)

Answer

600

Sanity-check the size. The roll has outer radius 2 in, so its outer circumference is only 2π ≈ 6.3 in — yet we got a length of about 628 in (over 52 ft!). Does 628 make sense? Yes: the tape is extremely thin (0.015 in), and roughly

\frac{2 - 1}{0.015}

≈ 67 layers stack between r = 1 and R = 2. Each layer is on average about 2π · 1.5 ≈ 9.4 in around, and 67 × 9.4 ≈ 630 in — matching 628 almost exactly. Good. Also 200π being our exact answer is a clean number, which is reassuring.

💡Key takeaway

This AMC 8 problem only needs Grade 7 circle-area π r² you already know — plus a clever picture that unrolls the tape into a thin rectangle!

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