AMC 8 · 2007 · #16

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

graph-readingarea-circlesperimeterpattern-recognition pattern-recognition ↑ Prerequisites: area-circlesperimeter

📏 Short solution 💡 2 insights 📊 Diagram

Problem

For each radius r = 1, 2, 3, 4, 5, Amanda plots the point (C, A), where C = 2π r is the circle's circumference and A = π r² is its area. Which of the five scatter plots could be her graph?

Pick an answer.

(A)

(C vs A scatter) five points with x equally spaced; y-values 2, 4, 7, 11, 16 — gaps grow 2, 3, 4, 5 (concave-up, quadratic-like)

(B)

(C vs A scatter) five points with x equally spaced; y-values 9, 6, 6, 9, 15 — dips then rises (non-monotonic)

(C)

(C vs A scatter) five points with x equally spaced; y-values 2, 6, 8, 6, 2 — rises then falls (inverted-U)

(D)

(C vs A scatter) five points with x equally spaced; y-values 2, 5, 8, 11, 14 — gaps all 3 (linear)

(E)

(C vs A scatter) five points with x equally spaced; y-values 15, 10, 6, 3, 1 — strictly decreasing

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Tool #5 (Look for a Pattern) is the right fit because the question is really asking how the area pattern behaves as the circumference grows steadily. Circumference grows by the same amount each time (+2π per unit of radius), but area grows by the differences of consecutive squares — 3, 5, 7, 9 — so the gaps between y-values must keep getting larger. Tool #2 (Make an Organized List) lets us tabulate the actual C and A values side by side so the growth pattern is visible. Once we see "equal x-steps, growing y-steps," only one choice can fit.

1STEP 1

Tabulate (C, A) for each radius using C = 2π r and A = π r².

r = 1: (C, A) = (2π, π); r = 2: (4π, 4π); r = 3: (6π, 9π); r = 4: (8π, 16π); r = 5: (10π, 25π)

2STEP 2

The C-values step up by 2π each time, so the five x-positions are evenly spaced.

Δ C = 2π between consecutive points → equal horizontal spacing

3STEP 3

The A-values π, 4π, 9π, 16π, 25π rise by 3π, 5π, 7π, 9π — the gaps keep growing.

Gaps in A: 4π-π=3π, 9π-4π=5π, 16π-9π=7π, 25π-16π=9π

4STEP 4

Only (A) fits: equal x-steps with y-gaps 2, 3, 4, 5 that keep growing — the quadratic shape.

(A) y-gaps: 4-2=2, 7-4=3, 11-7=4, 16-11=5 — strictly increasing →(A)

Answer

(C vs A scatter) five points with x equally spaced; y-values 2, 4, 7, 11, 16 — gaps grow 2, 3, 4, 5 (concave-up, quadratic-like)

Sanity check with π ≈ 3.14. Exact (C, A) values are approximately (6.3, 3.1), (12.6, 12.6), (18.8, 28.3), (25.1, 50.3), (31.4, 78.5). The A-values 3.1, 12.6, 28.3, 50.3, 78.5 accelerate upward — gaps 9.5, 15.7, 22.0, 28.3. That is the concave-up curve in choice (A), not the straight line in (D). Also, since A = C²/4π, plotting A against C traces a parabola opening upward — and (A) is the only choice whose five dots lie on such a curve.

💡Key takeaway

Circumference C = 2π r grows steadily, but area A = π r² grows by the odd numbers 3π, 5π, 7π, 9π — so the graph must rise faster and faster. That concave-up shape is choice (A).

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