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 "2011 AMC 10A Problems/Problem 14"

(→‎Solution)
(→‎Solution)
Line 13: Line 13:
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  
−
If <math>r<2</math> then the dice must show <math>(1,1),(1,2),(2,1)</math> which are <math>3</math> choices out of a total possible of <math>6 \times 6 =36</math>, so the probability is <math>3/36=1/12</math>
+
If <math>r<2</math> then the dice must show <math>(1,1),(1,2),(2,1)</math> which are <math>3</math> choices out of a total possible of <math>6 \times 6 =36</math>, so the probability is <math>\frac{3}{36}=\boxed{\frac{1}{12} \ \mathbf{(B)}}</math>.

Revision as of 00:13, 16 February 2011

Problem 14

A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

$\text{(A)}\,\frac{1}{36} \qquad\text{(B)}\,\frac{1}{12} \qquad\text{(C)}\,\frac{1}{6} \qquad\text{(D)}\,\frac{1}{4} \qquad\text{(E)}\,\frac{5}{18}$

Solution

We want the area, $\pi r^2$, to be less than the circumference, $2 \pi r$:

\begin{align*} \pi r^2 &< 2 \pi r \\ r &< 2 \end{align*}

If $r<2$ then the dice must show $(1,1),(1,2),(2,1)$ which are $3$ choices out of a total possible of $6 \times 6 =36$, so the probability is $\frac{3}{36}=\boxed{\frac{1}{12} \ \mathbf{(B)}}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10A_Problems/Problem_14&oldid=36972"