Difference between revisions of "2007 AMC 8 Problems/Problem 16"
m (Replaced content with "A is the answer.") |
m (Reverted edits by Raina0708 (talk) to last revision by Savannahsolver) (Tag: Rollback) |
||
(19 intermediate revisions by 11 users not shown) | |||
Line 1: | Line 1: | ||
− | A is the answer. | + | ==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 12:22, 16 August 2021
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?
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.