Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

2007 AMC 8 Problems/Problem 16: Difference between revisions

TwinDracula (talk | contribs)
editor
2 edits
Created page with 'A'
 
Imgreatatmath (talk | contribs)
editor
21 edits
 
(22 intermediate revisions by 13 users not shown)
Line 1: Line 1:
A
==Problem==
 
Amanda Reckonwith draws five circles with radii <math>1, 2, 3,
4</math> and <math>5</math>. Then for each circle she plots the point <math>(C,A)</math>,
where <math>C</math> is its circumference and <math>A</math> is its area. Which of the
following could be her graph?
 
<math> \textbf{(A)} </math>
<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>
 
<math> \textbf{(B)} </math>
<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>
 
<math> \textbf{(C)} </math>
<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>
 
<math> \textbf{(D)} </math>
<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>
 
<math> \textbf{(E)} </math>
<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 <math>2\pi</math>. 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 <math>\pi</math>. 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 <math>\boxed{\textbf{(A)}},
</math><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==
https://youtu.be/AW6BhCQ_ig8
 
~savannahsolver
 
==See Also==
{{AMC8 box|year=2007|num-b=15|num-a=17}}
{{MAA Notice}}

Latest revision as of 11:48, 24 December 2024

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

https://youtu.be/AW6BhCQ_ig8

~savannahsolver

See Also

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_16&oldid=237723"