Difference between revisions of "2007 AMC 8 Problems/Problem 16"
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size(75); | size(75); | ||
pair A= (1.5,2) , | pair A= (1.5,2) , | ||
| − | B= (3, | + | B= (3,4) , |
| − | C= (4.5, | + | C= (4.5,7) , |
| − | D= (6, | + | D= (6,11) , |
| − | E= (7.5, | + | E= (7.5,16) ; |
draw((0,-1)--(0,16)); | draw((0,-1)--(0,16)); | ||
draw((-1,0)--(16,0)); | draw((-1,0)--(16,0)); | ||
| Line 75: | Line 75: | ||
label("$A$", (0,8), W); | label("$A$", (0,8), W); | ||
label("$C$", (8,0), S);</asy> | 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== | ==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. 
