2007 AMC 8 Problems/Problem 16: Difference between revisions
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==Problem== | ==Problem== | ||
Amanda draws five circles with radii <math>1, 2, 3, | 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>, | 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 | where <math>C</math> is its circumference and <math>A</math> is its area. Which of the | ||
| Line 77: | Line 77: | ||
== Solution == | == 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> | 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== | ==See Also== | ||
{{AMC8 box|year=2007|num-b=15|num-a=17}} | {{AMC8 box|year=2007|num-b=15|num-a=17}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 11:48, 24 December 2024
Problem
Amanda Reckonwith draws five circles with radii 
and 
. Then for each circle she plots the point 
,
where 
is its circumference and 
is its area. Which of the
following could be her graph?
![[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]](http://latex.artofproblemsolving.com/5/4/4/5441a586e3c38f07d4057cb9867265ca14fdeb6e.png)
![[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]](http://latex.artofproblemsolving.com/2/3/0/230f178f3cd06089682b3fe4a6a9ecf9580ef492.png)
![[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]](http://latex.artofproblemsolving.com/9/5/d/95defaa77d8b1dd10283c438752810174f3e6d94.png)
![[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]](http://latex.artofproblemsolving.com/9/d/4/9d43329cbcfb1249b7fcf1fc15cd2b90bc666ef4.png)
![[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]](http://latex.artofproblemsolving.com/5/7/d/57d2b3b4be42360bf68db966abbc090187b5cdb3.png)
Solution
The circumference of a circle is obtained by simply multiplying the radius by 
. 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 
. 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 
.
-RBANDA
Video Solution by WhyMath
~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
