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

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

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.

@@ Line 17: / Line 17: @@
 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==

Page

Toolbox

Search