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2011 AMC 10A Problem 14

Problem 14 of 25IntermediateAlgebraGeometryProbability & Statistics

A pair of standard 66-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?

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Solution

For the area to be less than the circumference, we must have πr2<2πr \pi r^2 \lt 2\pi r r<2. r \lt 2. This means the diameter must be less than 4.4. There are three possible rolls that satisfy this: (1,1),(1,2),(2,1). (1, 1), (1, 2), (2, 1). The probability is then 336=112.\dfrac{3}{36} = \dfrac{1}{12}. Thus, B is the correct answer.
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Tagged: dice (probability) · circle area · inequality

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