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2006 AMC 8 Problems/Problem 7: Difference between revisions

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== Problem ==
== Problem 7 ==(  Jayesh is king  )


Circle <math> X </math> has a radius of <math> \pi </math>. Circle <math> Y </math> has a circumference of <math> 8 \pi </math>. Circle <math> Z </math> has an area of <math> 9 \pi </math>. List the circles in order from smallest to largest radius.  
 
Circle <math> X </math> has a radius of <math> \pi </math>. Circle <math> Y </math> has a circumference of <math> 8 \pi </math>. Circle <math> Z </math> has an area of <math> 9 \pi </math>. List the circles in order from smallest to the largest radius.  


<math> \textbf{(A)}\ X, Y, Z\qquad\textbf{(B)}\ Z, X, Y\qquad\textbf{(C)}\ Y, X, Z\qquad\textbf{(D)}\ Z, Y, X\qquad\textbf{(E)}\ X, Z, Y </math>
<math> \textbf{(A)}\ X, Y, Z\qquad\textbf{(B)}\ Z, X, Y\qquad\textbf{(C)}\ Y, X, Z\qquad\textbf{(D)}\ Z, Y, X\qquad\textbf{(E)}\ X, Z, Y </math>
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== Solution ==
== Solution ==


Using the formulas of circles, <math> C=2 \pi r </math> and <math> A= \pi r^2 </math>, we find that circle <math> Y </math> has a radius of <math> 4 </math> and circle <math> Z </math> has a radius of <math> 3 </math>. Thus, the order from smallest to largest radius is <math> \boxed{\textbf{(B)}\ Z, X, Y} </math>.
Not using the formulas of circles, <math> C=2 \pi r </math> and <math> A= \pi r^2 </math>, we find that circle <math> Y </math> has a radius of <math> 4 </math> and circle <math> Z </math> has a radius of <math> 3 </math>. Also, circle X has a radius of <math> \pi </math>. Thus, the order from smallest to largest radius is <math> \boxed{\textbf{(B)}\ Z, X, Y} </math>.
 
==Video Solution by OmegaLearn==
https://youtu.be/J5-hDWd28tM


==See Also==
==See Also==
{{AMC8 box|year=2006|num-b=6|num-a=8}}
{{AMC8 box|year=2006|num-b=6|num-a=8}}
{{MAA Notice}}

Latest revision as of 21:26, 26 October 2025

== Problem 7 ==( Jayesh is king )


Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to the largest radius.

$\textbf{(A)}\ X, Y, Z\qquad\textbf{(B)}\ Z, X, Y\qquad\textbf{(C)}\ Y, X, Z\qquad\textbf{(D)}\ Z, Y, X\qquad\textbf{(E)}\ X, Z, Y$

Solution

Not using the formulas of circles, $C=2 \pi r$ and $A= \pi r^2$, we find that circle $Y$ has a radius of $4$ and circle $Z$ has a radius of $3$. Also, circle X has a radius of $\pi$. Thus, the order from smallest to largest radius is $\boxed{\textbf{(B)}\ Z, X, Y}$.

Video Solution by OmegaLearn

https://youtu.be/J5-hDWd28tM

See Also

2006 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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: File missing

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