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Difference between revisions of "2007 AMC 8 Problems/Problem 16"

(Solution to Problem 16)
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<math>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?</math>
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==Problem==
  
 +
Amanda Reckonwith draws five circles with radii <math>1, 2, 3,
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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>
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<asy>
 +
size(75);
 +
pair A= (1.5,2) ,
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B= (3,4) ,
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C= (4.5,7) ,
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D= (6,11) ,
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E= (7.5,16) ;
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draw((0,-1)--(0,16));
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draw((-1,0)--(16,0));
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dot(A^^B^^C^^D^^E);
 +
label("$A$", (0,8), W);
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label("$C$", (8,0), S);</asy>
  
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Go here for the diagram:
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<math> \textbf{(B)} </math>
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[[Media:http://www.artofproblemsolving.com/Wiki/index.php/File:AMC8_2007_16.png]]
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<asy>
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size(75);
 +
pair A= (1.5,9) ,
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B= (3,6) ,
 +
C= (4.5,6) ,
 +
D= (6,9) ,
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E= (7.5,15) ;
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draw((0,-1)--(0,16));
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draw((-1,0)--(16,0));
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dot(A^^B^^C^^D^^E);
 +
label("$A$", (0,8), W);
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label("$C$", (8,0), S);</asy>
  
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The answer is A 
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<math> \textbf{(C)} </math>
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If you look at the diagrams, you can see the center in A where the points are at 1,1  2,2  3,3, etc.
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<asy>
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Thus, its's the answer.
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size(75);
 +
pair A= (1.5,2) ,
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B= (3,6) ,
 +
C= (4.5,8) ,
 +
D= (6,6) ,
 +
E= (7.5,2) ;
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draw((0,-1)--(0,16));
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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) ,
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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

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

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