# 2007 AMC 8 Problems/Problem 16

## 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? $\textbf{(A)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,4) , C= (4.5,7) , D= (6,11) , E= (7.5,16) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$ $\textbf{(B)}$ $[asy] size(75); pair A= (1.5,9) , B= (3,6) , C= (4.5,6) , D= (6,9) , E= (7.5,15) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$ $\textbf{(C)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,6) , C= (4.5,8) , D= (6,6) , E= (7.5,2) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$ $\textbf{(D)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,5) , C= (4.5,8) , D= (6,11) , E= (7.5,14) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$ $\textbf{(E)}$ $[asy] size(75); pair A= (1.5,15) , B= (3,10) , C= (4.5,6) , D= (6,3) , E= (7.5,1) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$

## Solution

The circumference of a circle is obtained by simply multiplying the radius by $2\pi$. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by $\pi$. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is $\boxed{\textbf{(A)}},$ $[asy] size(75); pair A= (1.5,2) , B= (3,4) , C= (4.5,7) , D= (6,11) , E= (7.5,16) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("A", (0,8), W); label("C", (8,0), S);[/asy]$. -RBANDA

## Video Solution by WhyMath

~savannahsolver

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 