2011 AMC 12A Problems/Problem 10
Problem
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?
Solution
It is clear that only a diameter of and would result in the circumference being larger than the radius.
For the radius is so The area is $\pi r^2 \rightarrow \pi 1^2 \right arrow \pi$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) Thus, so we need snake eyes or and the probability is
By the same work using diameter of , we find that the circumference is greater than the area. So would be found by rolling a and a so the probability is .
Thus, those are the only two cases however there are ways to roll a with die so the probability is
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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All AMC 12 Problems and Solutions |