Question

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?

$\dfrac{1}{36}$

$\dfrac{1}{12}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{5}{18}$

Answer: B

Concepts:dice (probability)circle area inequality

Difficulty rating: 1140

Solution:

For the area to be less than the circumference, we must have $\pi r^2 \lt 2\pi r$ $r \lt 2.$

This means the diameter must be less than $4.$ There are three possible rolls that satisfy this: $(1, 1), (1, 2), (2, 1).$

The probability is then $\dfrac{3}{36} = \dfrac{1}{12}.$

Thus, B is the correct answer.

2011 AMC 10A Problem 14

Answer: B

Difficulty rating: 1140

Solution:

Problem 14 in Other Years