Difference between revisions of "2011 AMC 12A Problems/Problem 10"

(Problem)
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== Solution ==
 
== Solution ==
 
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For the circumference to be greater than the area, we must have <math>\pi d > \pi (\frac{d}{2})^2</math>, or <math>d<4</math>. Now since <math>d</math> is determined by a sum of two dice, the only possibilities for <math>d</math> are thus <math>2</math> and <math>3</math>. In order for two dice to sum to <math>2</math>, they most both show a value of <math>1</math>. The probability of this happening is <math>\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}</math>. In order for two dice to sum to <math>3</math>, one must show a <math>1</math> and the other must show a <math>2</math>. Since this can happen in two ways, the probability of this event occurring is <math>2 \times \frac{1}{6} \times \frac{1}{6} = \frac{2}{36}</math>. The sum of these two probabilities now gives the final answer: <math>\frac{1}{36} + \frac{2}{36} = \frac{3}{36} = \frac{1}{12} \rightarrow \boxed{textbf{B}}</math>
It is clear that only a diameter of <math>2</math> and <math>3</math> would result in the circumference being larger than the radius.  
 
 
 
For <math>2,</math> the radius is <math>1</math> so <math>2\pi r \rightarrow 2\pi(1) \rightarrow 2\pi </math> The area is <math> \pi r^2 \rightarrow \pi 1^2 \rightarrow \pi </math> Thus, <math> 2\pi > \pi </math> so we need snake eyes or <math>2</math> <math>1's</math> and the probability is <math> \dfrac{1}{6} \cdot  \dfrac{1}{6} \rightarrow \dfrac{1}{36} </math>
 
 
 
By the same work using diameter of <math>3</math>, we find that the circumference is greater than the area. So <math>3</math> would be found by rolling a <math>1</math> and a <math>2</math> so the probability is <math> \dfrac{1}{6} \cdot \dfrac{1}{6} \rightarrow \dfrac{1}{36} </math>.  
 
 
 
Thus, those are the only two cases however there are <math>2</math> ways to roll a <math>3</math> with <math>2</math> die so the probability is  <math> \dfrac{1}{36} + \dfrac{1}{36} + \dfrac{1}{36} \rightarrow \dfrac{3}{36} \rightarrow \dfrac{1}{12} \rightarrow \boxed{B} </math>
 
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=9|num-a=11|ab=A}}
 
{{AMC12 box|year=2011|num-b=9|num-a=11|ab=A}}

Revision as of 20:23, 10 February 2011

Problem

A pair of standard $6$-sided 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?

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

Solution

For the circumference to be greater than the area, we must have $\pi d > \pi (\frac{d}{2})^2$, or $d<4$. Now since $d$ is determined by a sum of two dice, the only possibilities for $d$ are thus $2$ and $3$. In order for two dice to sum to $2$, they most both show a value of $1$. The probability of this happening is $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$. In order for two dice to sum to $3$, one must show a $1$ and the other must show a $2$. Since this can happen in two ways, the probability of this event occurring is $2 \times \frac{1}{6} \times \frac{1}{6} = \frac{2}{36}$. The sum of these two probabilities now gives the final answer: $\frac{1}{36} + \frac{2}{36} = \frac{3}{36} = \frac{1}{12} \rightarrow \boxed{textbf{B}}$

See also

2011 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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 12 Problems and Solutions
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