2011 AMC 10A Problems/Problem 14

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 $3/36=1/12$

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2011 AMC 10A Problems/Problem 14

Problem 14

Solution