AMC 8 · 2007 · #16

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search

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

📏 Short solution 💡 2 insights 📊 Diagram

Problem

Amanda Reckonwith draws five circles with radii $1, 2, 3, 4$ and $5$ . Then for each circle she plots the point $(C,A)$ ,
where $C$ is its circumference and $A$ is its area. Which of the
following 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

View mode:

Toolkit + CCSS Solution

Understand

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

Givens: Five radii: $r = 1, 2, 3, 4, 5$; $x$-coordinate of each point is $C = 2\pi r$; $y$-coordinate of each point is $A = \pi r^2$; Answer choices: (A) y-values $2, 4, 7, 11, 16$ — gaps grow; (B) y-values $9, 6, 6, 9, 15$ — dips then rises; (C) y-values $2, 6, 8, 6, 2$ — inverted-U; (D) y-values $2, 5, 8, 11, 14$ — linear; (E) y-values $15, 10, 6, 3, 1$ — strictly decreasing

Unknowns: Which scatter plot matches the actual $(C, A)$ values for the five circles

Understand

Plan

Primary tool: #5 Look for a Pattern

Secondary: #2 Make an Organized List

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 ($+2\pi$ 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.

Execute — Answer: A

#2 Make an Organized List 7.G.B.4 Step 1

List $C$ and $A$ for each radius using $C = 2\pi r$ and $A = \pi r^2$.
Keep $\pi$ as a common factor so the pattern is easy to read.

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

💡 The Grade 7 circle formulas plug in directly: $C$ is linear in $r$, $A$ is the square of $r$ (times $\pi$).

#5 Look for a Pattern 6.RP.A.1 Step 2

Check the $x$-coordinates.
The $C$-values $2\pi, 4\pi, 6\pi, 8\pi, 10\pi$ are equally spaced — each one is $2\pi$ more than the previous.
So the five points must sit at evenly spaced $x$-positions on the graph.

$\Delta C = 2\pi$ between consecutive points $\Rightarrow$ equal horizontal spacing

💡 Doubling, tripling, ... the radius doubles, triples, ... the circumference. Steady step.

#5 Look for a Pattern 6.EE.A.1 Step 3

Check the $y$-coordinates.
The $A$-values $\pi, 4\pi, 9\pi, 16\pi, 25\pi$ are the perfect squares times $\pi$.
The jumps between them are $3\pi, 5\pi, 7\pi, 9\pi$ — each gap is larger than the last.
So the points rise, AND they rise by more each time (concave-up shape).

Gaps in $A$: $4\pi-\pi=3\pi,\ 9\pi-4\pi=5\pi,\ 16\pi-9\pi=7\pi,\ 25\pi-16\pi=9\pi$

💡 Consecutive square differences are the odd numbers $3, 5, 7, 9$ — a classic growing pattern.

#5 Look for a Pattern 8.F.A.3 Step 4

Match the pattern "equal $x$-spacing, growing $y$-gaps" to the choices.
(B), (C), (E) fail because their $y$-values are not strictly increasing.
(D) has $y$-gaps $3, 3, 3, 3$ — constant, which would mean $A$ is linear in $C$, but $A$ grows like $r^2$, not like $r$.
Only (A) has $y$-gaps $2, 3, 4, 5$ — strictly growing, exactly the concave-up shape forced by squaring.

(A) $y$-gaps: $4-2=2,\ 7-4=3,\ 11-7=4,\ 16-11=5$ — strictly increasing $\Rightarrow\textbf{(A)}$

💡 A linear graph has equal step-ups; a quadratic graph has step-ups that keep growing. (A) shows the quadratic shape.

[1] #2 7.G.B.4 List $C$ and $A$ for each radius using $C = 2\pi r$ and $A = \pi r^2$. Keep $\pi

[2] #5 6.RP.A.1 Check the $x$-coordinates. The $C$-values $2\pi, 4\pi, 6\pi, 8\pi, 10\pi$ are eq

[3] #5 6.EE.A.1 Check the $y$-coordinates. The $A$-values $\pi, 4\pi, 9\pi, 16\pi, 25\pi$ are th

[4] #5 8.F.A.3 Match the pattern "equal $x$-spacing, growing $y$-gaps" to the choices. (B), (C)

Review

Reasonableness: Sanity check with $\pi \approx 3.14$. 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 $A = \dfrac{C^2}{4\pi}$, plotting $A$ against $C$ traces a parabola opening upward — and (A) is the only choice whose five dots lie on such a curve.

Alternative: Tool #10 (Find a Related Problem): substitute $r = \dfrac{C}{2\pi}$ into $A = \pi r^2$. That gives $A = \pi \cdot \dfrac{C^2}{4\pi^2} = \dfrac{C^2}{4\pi}$, the equation of a parabola in $C$. A parabola through the origin opening upward is concave up and strictly increasing on the right side, instantly matching choice (A) without computing each point.

CCSS standards used (min grade 8)

7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Computing $C = 2\pi r$ and $A = \pi r^2$ for each radius $r = 1, 2, 3, 4, 5$.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Noticing that $C$ grows in equal $2\pi$ steps, so the $x$-coordinates are evenly spaced.)
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Evaluating the squares $1^2, 2^2, 3^2, 4^2, 5^2$ to get the $A$-values $\pi, 4\pi, 9\pi, 16\pi, 25\pi$.)
8.F.A.3 Interpret the equation $y = mx + b$ as defining a linear function; give examples of functions that are not linear (Distinguishing the quadratic growth pattern of $A$ (gaps $3\pi, 5\pi, 7\pi, 9\pi$) from a linear graph with constant gaps, so the correct choice (A) is the non-linear one.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 8)

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