🇺🇸 AMC 8 ⇄ switch contest
2007 AMC 8

Problem 16

Problem 16 · 2007 AMC 8 Medium
Algebra & Patterns quadratic-vs-linear

Amanda 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?

Figure for AMC 8 2007 Problem 16
Show answer
Answer: A — Graph A.
Show hints
Hint 1 of 2
You don't need actual numbers — just ask how area grows compared to circumference. Circumference is linear in r, but area is squared, so as C climbs steadily, A should shoot up faster and faster.
Still stuck? Show hint 2 →
Hint 2 of 2
Eliminate the radius to see the relationship: with C = 2πr and A = πr2, A = C2/(4π) — a curve that bends upward, not a straight line.
Show solution
Approach: decide the shape of the A-vs-C curve
  1. Substitute r = C/(2π) into A = πr2 to get A = C2/(4π). So A is proportional to C squared.
  2. That's a curve through the origin that rises and steepens — concave up — never a straight line. Only graph A bends that way.
  3. Mental check: plot the smallest and a bigger circle: (2π, π) then (10π, 25π). The y-values blow up much faster than the x-values — the hallmark of a parabola, which rules out every straight-line option.
Mark: · log in to save
↑