Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2010 AMC 10A Problems/Problem 5"

(Created page with '144')
 
(→‎Solution)
 
(5 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
144
+
== Problem 5 ==
 +
The area of a circle whose circumference is <math>24\pi</math> is <math>k\pi</math>. What is the value of <math>k</math>?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 6
 +
\qquad
 +
\mathrm{(B)}\ 12
 +
\qquad
 +
\mathrm{(C)}\ 24
 +
\qquad
 +
\mathrm{(D)}\ 36
 +
\qquad
 +
\mathrm{(E)}\ 144
 +
</math>
 +
 
 +
==Solution==
 +
 
 +
If the circumference of a circle is <math>24\pi</math>, the radius would be <math>12</math>. Since the area of a circle is <math>\pi r^2</math>, the area is <math>144\pi</math>. The answer is <math>\boxed{E}</math>.
 +
 
 +
 
 +
==Solution 2==
 +
By definition, <math>\pi</math> is the ratio of the circumference to the diameter. Since the circumference is <math>24\pi</math>, the diameter must be <math>24</math> and the radius is <math>12</math>. Therefore, by the area of circle formula <math>A=\pi r^{2}</math> the area is <math>12^{2}\pi=144\pi</math> and <math>k=144 \Longrightarrow \boxed{\textbf{(E)} 144}</math>.
 +
 
 +
==Video Solution==
 +
https://youtu.be/C1VCk_9A2KE?t=290
 +
 
 +
~IceMatrix
 +
 
 +
== See Also ==
 +
 
 +
{{AMC10 box|year=2010|ab=A|num-b=4|num-a=6}}
 +
{{MAA Notice}}

Latest revision as of 05:12, 29 June 2022

Problem 5

The area of a circle whose circumference is $24\pi$ is $k\pi$. What is the value of $k$?

$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 24 \qquad \mathrm{(D)}\ 36 \qquad \mathrm{(E)}\ 144$

Solution

If the circumference of a circle is $24\pi$, the radius would be $12$. Since the area of a circle is $\pi r^2$, the area is $144\pi$. The answer is $\boxed{E}$.


Solution 2

By definition, $\pi$ is the ratio of the circumference to the diameter. Since the circumference is $24\pi$, the diameter must be $24$ and the radius is $12$. Therefore, by the area of circle formula $A=\pi r^{2}$ the area is $12^{2}\pi=144\pi$ and $k=144 \Longrightarrow \boxed{\textbf{(E)} 144}$.

Video Solution

https://youtu.be/C1VCk_9A2KE?t=290

~IceMatrix

See Also

2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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 AMC 10 Problems and Solutions

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_10A_Problems/Problem_5&oldid=175449"